The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
281,179 characters
Testing Sign Agreement
\maketitle
\vspace{-10pt}
\begin{spacing}{1.1}
\begin{abstract}
This article considers the problem of testing sign agreement among a finite number of parameters. This problem arises in empirical settings such as detecting treatment effects with opposite signs across subgroups, outcomes, or time periods, and testing instrument validity for local average treatment effects. For the null hypothesis that the parameters are either all non-negative or all non-positive, I propose two novel tests: a least favorable test and a conditional test. The least favorable test uses a worst-case null critical value, while the conditional test first screens components with large positive or negative estimates and then tests the remaining sign-unresolved components conditional on the screening event. Unlike existing tests, both procedures accommodate arbitrary dependence among estimators; in the special case of independent estimators, the critical values depend only on the dimension and testing levels. We show that both tests control asymptotic size uniformly over a large class of nonparametric distributions. Local asymptotic power analysis reveals a tradeoff: the least favorable test is more powerful near boundary configurations where sign restrictions bind, whereas the conditional test is more powerful when some components are well separated from zero. Simulation evidence supports these theoretical predictions in finite samples.
\end{abstract}
\end{spacing}
\noindent\textit{KEYWORDS: union of moment inequalities, uniform asymptotic validity, bootstrap}\\
\noindent\textit{JEL classification codes: C12, C31, C35, C36}
\section{Introduction}
This article studies the problem of testing whether a finite collection of parameters share a common sign. Writing
$\mu=(\mu_1,\ldots,\mu_k)$ for the vector of parameters of interest, which may
include regular estimands such as OLS, instrumental-variable, two-way
fixed-effects, or difference-in-differences coefficients, the hypothesis of \emph{sign agreement} is
\begin{align}\label{def:intro-null}
H_0:\ \mu_j\geq 0 \text{ for all } j=1,\ldots,k
\quad\text{or}\quad
\mu_j\leq 0 \text{ for all } j=1,\ldots,k,
\end{align}
tested against the alternative that the parameters disagree in sign:
\begin{align*}
H_1:\ \text{there exist } i\neq j \text{ such that }
\mu_i>0 \text{ and } \mu_j<0.
\end{align*}
Rejection is direct evidence that the $k$ parameters do not all point the same way---that some are positive while others are negative.
The hypothesis in \eqref{def:intro-null} arises across many empirical settings. When the $k$ coordinates index post-treatment horizons, sign agreement asks whether the dynamic response reverses over event time, separating effects that merely strengthen or attenuate from those whose short-run and long-run signs differ: immediate gains that are later offset, or short-run costs that yield delayed benefits. When the coordinates instead index subgroups of a treated population, sign agreement is the absence of what the biostatistics literature calls \emph{qualitative interaction}---the treatment helping one group while
harming another. It also underlies a testable implication of instrument validity for the local average treatment effect \citep{ImbensAngrist1994ECTA-LATE,Kitagawa2021JoE}. For an ordered outcome, it characterizes whether two distributions are ordered by first-order stochastic dominance. In the labor-market discrimination model of \citet{Bharadwaj/Deb/Renou:2024}, dominance between the outcome distributions of two groups is the configuration consistent with taste-based discrimination, so a rejection establishes that the distributions cross and points instead to statistical or no discrimination. Section~\ref{sec:examples} develops these examples.
A feature common to these settings is that the $k$ estimators are typically correlated---across horizons or by construction when a discretized outcome yields multinomial cell frequencies. Despite this breadth of applications, existing procedures developed for testing sign agreement remain confined to restrictive settings. Developed primarily in biostatistics as qualitative interaction tests, most procedures assume that the components of the underlying data vector are statistically independent (\citealt{GailSimon1985Biometrics}, \citealt{Silvapulle2001Biometrics},
\citealt{PiantadosiGail1993}, \citealt{LiChan2006detecting}, and
\citealt*{Zhao2019JASA}). The test of \cite{RussekSimon1993Biometrics}
permits arbitrary correlation, but only when $k=2$. The recent procedures in economics of \citet*{BrinchMogstadWiswall2017JPE}, \cite{kowalski2023RES}, and \citet*{MillerMolinariStoye2024WP} likewise treat the two-dimensional case.
These restrictions rule out leading applications---for instance, detecting
opposite treatment effects across several correlated outcomes.
We develop two tests of sign agreement---a least favorable test and a
conditional test---that accommodate an arbitrary correlation structure among the $k$ estimators and any finite $k$. Our main result establishes that both are uniformly asymptotically valid over a large class of nonparametric distributions under a standard uniform integrability condition: their finite-sample size is controlled at the nominal level in large samples. The uniform asymptotic validity is stronger than the pointwise asymptotic validity, which is a common requirement for tests. Because the asymptotic distribution of the test statistic is discontinuous in the underlying parameters, pointwise asymptotics can understate size in finite samples \citep{mikusheva2007uniform,AndrewsGuggenberger2009ECTA-hybrid,
AndrewsGuggenberger2010ET}. Both tests are consistent against every fixed alternative and we compare them through their local asymptotic power. Neither dominates: the least favorable test is more powerful near boundary configurations where the sign restriction binds, whereas the conditional test is more powerful when several components are well separated from zero. To our knowledge, these are the first uniform validity results for tests of sign agreement over a nonparametric class; existing results establish finite-sample validity only within restricted families of normal distributions, or pointwise asymptotic validity.
A challenge in building a uniformly asymptotically valid sign agreement test is that, under the composite null, the limiting distribution of the test statistic is not pinned down and depends on a nuisance parameter that cannot be estimated uniformly over the null. A critical value based on the distribution at any single value of the nuisance parameter would fail to control size uniformly. To circumvent this, we adopt the least favorable approach, taking the largest critical value attainable over all nuisance configurations consistent with the null, while keeping the test nonconservative. We implement this approach with three test statistics---a maximum, a sum of squares, and a quasi-likelihood ratio---which differ in how they measure the extent of sign disagreement. Critical values are computed by parametric bootstrap, and a nonparametric bootstrap version is also available. The resulting family of tests subsumes the tests of \citet{GailSimon1985Biometrics}
and \citet{PiantadosiGail1993} under independent estimators, and that of
\citet{RussekSimon1993Biometrics} with two parameters.
The conditional test is designed to improve the power against a wide range of alternatives by exploiting the observed signs of the estimates. It proceeds in two steps. Step~I screens each coordinate, classifying it as strong evidence of a positive estimate, strong evidence of a negative estimate, or sign-unresolved, using a threshold calibrated at a tuning level $\tau\in(0,\alpha)$. Step~II applies a contingent rule: it does not reject when all screened coordinates point the same way, rejects when both signs appear, and otherwise tests only the sign-unresolved coordinates---with a one-sided test when some signs are resolved, or a conditional version of the least favorable test when none are. The test is \emph{conditional} because this second-step critical value is computed from the distribution of the statistic conditional on the Step~I screening event, so both the statistic and its critical value adapt to what the data reveal. When the estimators are independent, these conditional critical values are available in tabulated form, requiring no bootstrap or numerical optimization. Screening is what drives the comparison. When several estimates carry strong directional evidence, screening removes them and the second step effectively operates in a lower dimension; the conditional test then overtakes the least favorable test. When the estimates sit near the boundary and little is screened, the two nearly coincide and the least favorable test's larger effective level prevails.
The construction of the conditional test is closely related to ideas from
selective inference. Its screening-then-conditioning logic parallels selective inference, in which a statistic is evaluated conditional on a data-dependent selection event: the rectangular screening region of Step~I is such an event, and we exploit the truncated-Gaussian structure it induces \citep{lee2016AoS-post-selection-LASSO}. The problems differ, however: selective inference typically concerns a parameter or model selected using the data, whereas our null hypothesis is fixed in advance and screening is used only to sharpen the subsequent test.
The paper is also related to the literature on one-sided moment inequalities,
where information about the signs or slackness of moments is used to improve
upon worst-case inference, including generalized moment selection and related
refinements \citep*{Hansen2005JBES,AndrewsSoares2010ECTA,bugni2010ECTA,
Canay2010JoE,andrewsbarwick2012ecta,RomanoShaikhWolf2014ECTA,
chernozhukov2019RES}. Since the sign-agreement null is the union of the two
one-sided hypotheses, one-sided moment-inequality procedures can also be applied in both directions and combined through the intersection--union principle to obtain generic tests of sign agreement; see Remark~\ref{rem:LF-IUT-comparison}. The tests developed here instead exploit the union structure directly. The LF test directly calibrates the test statistic over the union of the two orthants, while the conditional test additionally uses data-dependent directional screening to adapt both the statistic and the critical value to the realized configuration. This distinction is important because, unlike in a one-sided moment-inequality problem, the distribution of our sign-agreement statistic is not stochastically monotone in the nuisance mean vector, so the
usual moment-selection refinements do not directly carry over to our setting.
A simulation study supports these theoretical findings. Across $k=3,5,$ and $8$ and the three correlation designs, the empirical maximum null rejection probabilities of the LF and conditional tests remain close to the nominal significance level. The power comparison follows the pattern predicted by the theoretical results: when only one coordinate provides strong directional information, the two proposed tests have similar power, whereas the conditional test increasingly outperforms the LF test as more coordinates become well separated from zero. We also compare the proposed tests with intersection--union adaptations of the one-sided moment-inequality procedures of \citet{Cox/Shi:2023} and \citet*{RomanoShaikhWolf2014ECTA}. Both proposed tests are competitive with these benchmarks. The test by \citet{Cox/Shi:2023} can be more powerful at moderate signal strengths but is often overtaken as the signals strengthen, whereas the test by \citet*{RomanoShaikhWolf2014ECTA} is generally less powerful except under some of the strongest dense alternatives. Overall, no
procedure is uniformly more powerful across the alternatives considered.
We illustrate the proposed tests in two applications, whose parameter estimators are correlated. First, we revisit the dynamic effect of unilateral divorce laws on divorce rates \citep{Wolfers2006AER}. Testing sign agreement across eight post-reform horizons asks whether the response reverses sign over event time; the tests reject under the fixed-effects specification and find suggestive evidence of sign reversal. Second, we test the validity of the college-proximity instrument of \citet{card1993NBER} for the returns to college---the assumptions under which proximity to a college identifies a causal return. Discretizing the outcome casts instrument validity as sign agreement. The tests reject at the $1\%$ level, implying that the two-stage least squares estimand identifies neither the complier nor the defier average effect.
The remainder of the article is organized as follows.
Section~\ref{sec:examples} presents motivating examples.
Section~\ref{sec:setup} fixes notation and defines uniform asymptotic validity.
Section~\ref{sec:LFCtest} develops the least favorable test, and
Section~\ref{sec:conditional-test} the conditional test, together with their
uniform validity results. Section \ref{sec:power-comparison} provides power results. Section~\ref{sec:MonteCarlo-Simulation} reports the
simulation study and Section~\ref{sec:Empirical-Application} the two empirical
illustrations. Section~\ref{sec:Conclusion} concludes. Proofs and further
results are collected in the appendices.
\section{Motivating Examples}\label{sec:examples}
This section presents empirical contexts where the sign agreement hypothesis emerges naturally.
\begin{example}[Sign-reversing Dynamic Treatment Effect]
\rm\label{example:Heterogeneity-RCT}
A treatment effect may reverse sign over time, so that its short-run and
long-run consequences point in opposite directions. The sign agreement
hypothesis provides a framework for examining whether such reversal occurs.
To be concrete, consider a setting with a binary treatment. Let $D \in \{0,1\}$
denote treatment status, with $D=1$ for treated individuals, and let $Y_{1}$
and $Y_{0}$ denote the potential outcomes with and without treatment. Suppose
outcomes are observed at $k$ post-treatment horizons $g_1,\ldots,g_k$, and let
$$\mu_j=\mathbb{E}[Y_{1}(g_j) - Y_{0}(g_j)] \qquad j=1,\ldots,k$$
denote the average treatment effect at horizon $g_j$. The hypothesis in
\eqref{def:intro-null} then states that the dynamic response does not reverse
sign over event time. It distinguishes treatments whose effects merely
strengthen or attenuate from those whose short-run and long-run effects have
opposite signs---a distinction that is often economically substantive, since an
intervention may generate immediate gains that are subsequently offset, or
impose short-run costs that produce delayed benefits. The empirical
illustration in Section~\ref{sec:Empirical-Application} provides a
quasi-experimental application.
The same hypothesis applies across subgroups rather than horizons. Suppose each
individual belongs to one of $k$ groups indexed by an observable discrete
variable $G \in \mathcal{G} \equiv \{g_1, \dots, g_k\}$, and that under full
compliance the average treatment effect for group $g_j$ is
$$\mu_j=\mathbb{E}[Y_{1} - Y_{0} \mid G = g_j] \qquad j=1,\ldots,k.$$
Then \eqref{def:intro-null} states that the group-level effects share a common
sign, so that treatment does not help one group while harming another---the
absence of what the biostatistics literature calls \textit{qualitative
interaction}. Hypotheses of this form arise routinely in clinical trials; see,
among others, \citet{rastogi2008preoperative}, \citet{NEJM2022},
\citet{Jama2019}, and \citet{Lancet2019}.
\end{example}
\begin{example}[Instrument Validity for Local Average Treatment Effect]\label{example:LATE-instrument-validity}\rm
\cite{ImbensAngrist1994ECTA-LATE} demonstrate that under certain assumptions, the canonical two-stage least squares (TSLS) estimand has a causal interpretation as a local average treatment effect (LATE). The sign agreement hypothesis provides a framework to test an implication of these assumptions, particularly when the outcome is discrete. Let \(Y\in\mathcal{Y}\), \(D\in{0,1}\), and \(Z\in{0,1}\) denote the observed outcome, treatment status, and instrument, respectively. Let $Y_d$ denote the potential outcome under treatment status $d$, satisfying $Y=Y_1D+Y_0(1-D).$ Let $D_z$ denote the potential treatment status under instrument value $z$. These potential treatment responses define subpopulations: \textit{compliers} with $(D_0, D_1)=(0,1)$ and \textit{defiers} with $(D_0, D_1)=(1,0)$. Given a nonzero first stage, the LATE interpretation of the TSLS estimand relies on the following assumptions:
\begin{align*}
\begin{aligned}
&\text{Exogeneity: }(D_0, D_1, Y_{1}, Y_{0}) \text{ and } Z \text{ are independent and }\\
&\text{Monotonicity: }D_1 \geq D_0 \text{ or }D_1 \leq D_0\text { a.s.}
\end{aligned}
\end{align*}
Under \(D_1\geq D_0\), defiers are absent and the TSLS estimand equals the average treatment effect among compliers. Under \(D_1\leq D_0\), compliers are absent and the estimand equals the average treatment effect among defiers. These assumptions are not directly testable, but they can be refuted through a testable implication. Suppose that the outcome is discretized: let $\mathcal{Y}_1,\ldots,\mathcal{Y}_k$
partition the support of $Y$ into $k$ categories, which may be the support
points of a discrete outcome or bins of a continuous one. Define
\begin{align}\label{eq:instrument-validity-parameters}
\mu_j&=P\{Y\in\mathcal{Y}_j,\, D=1\,|\,Z=1\}- P\{Y\in\mathcal{Y}_j,\, D=1\,|\,Z=0\} \qquad j=1,...,k\\
\mu_{k+j}&=P\{Y\in\mathcal{Y}_j,\, D=0\,|\,Z=0\}- P\{Y\in\mathcal{Y}_j,\, D=0\,|\,Z=1\} \qquad j=1,...,k.
\end{align}
The testable implication can then be formulated analogously to \eqref{def:intro-null}:
\begin{align}\label{eq:instrument-validity-null}
H_0: \mu_j\geq 0 \quad \text{for all }j=1,...,2k \quad\text{or}\quad\mu_j\leq 0 \quad \text{for all }j=1,...,2k;
\end{align}
see \cite{Kitagawa2021JoE}. Because the underlying implication holds for every Borel set,
\eqref{eq:instrument-validity-null} is a valid implication for any choice of partition, and it is exact for discrete $Y$ when the categories are the support points. Rejecting the hypothesis refutes the joint assumptions, implying that the TSLS estimand does not admit the causal interpretation as LATE.\footnote{Several procedures are available for testing instrument validity under the conventional one-sided monotonicity condition $D_1 \geq D_0$, which rules out defiers (e.g., \citealt{Kitagawa2015ECTA}; \citealt{Huber2015REStat-testingLATE}; \citealt{Mourifie2017REStat-testingLATE}; and \citealt*{Machado2019JoE}). These procedures are appropriate when researchers can establish a priori that compliers exist and therefore need only assess the no-defiers restriction. In contrast, we consider the two-sided monotonicity condition, under which either $D_1 \geq D_0$ or $D_0 \geq D_1$; that is, either defiers or compliers are absent.}
\end{example}
\begin{example}[Statistical versus Taste-Based Discrimination] \rm\label{example:discrimination}
Economists distinguish between two forms of labor-market discrimination. Statistical discrimination arises when employers imperfectly observe individual productivity and therefore rely on group-specific signals, whereas taste-based discrimination reflects preferences or animus toward a particular group \citep{Becker1957,Phelps1972}. \citet{Bharadwaj/Deb/Renou:2024} show that, when two groups have the same mean productivity, their observed wage distributions \(G_1\) and \(G_2\) can be rationalized without taste-based discrimination if and only if neither distribution strictly first-order stochastically dominates the other. Strict stochastic dominance therefore provides evidence of taste-based discrimination, whereas the absence of dominance is consistent with statistical discrimination. Suppose that the outcome is discrete with ordered support $w_1<\cdots<w_J$ where $J\geq 3,$ as may occur when wages are reported in intervals or when the outcome is an ordered non-wage measure, such as promotion rank. Define
\begin{align*}
\mu_j=G_1(w_j)-G_2(w_j),
\qquad j=1,\ldots,J-1.
\end{align*}
First-order stochastic dominance can then be characterized by the signs of these differences. The two distributions are weakly ordered by first-order stochastic dominance if and only if the components of \(\mu\) have a common weak sign. Applying the sign-agreement hypothesis in \eqref{def:null-hypo} to \(\mu\) therefore yields a test of the null that the wage distributions are ordered by first-order stochastic dominance—the configuration that, under the maintained assumptions of \citet{Bharadwaj/Deb/Renou:2024}, includes taste-based discrimination. Rejecting the null establishes that the distributions cross and hence that the observed wage distributions are consistent with statistical discrimination or no discrimination. Failure to reject leaves the dominance configuration possible but does not establish either dominance or taste-based discrimination.
\end{example}
Beyond these three applications, sign agreement is relevant in other settings. One arises when a common underlying outcome is captured by multiple measures: rather than aggregating them into an index, researchers may examine whether the corresponding treatment effects have a common sign. The framework can also be used to assess sign agreement across study-specific effects in meta-analysis \citep[e.g.,][]{Zhao2019JASA,SloughTyson2024}. \cite*{MillerMolinariStoye2024WP} discuss further examples involving two target parameters.
A common feature of these examples is that the estimators of $\mu_1,...,\mu_k$ can be correlated. Independence is plausible only when the components are estimated from disjoint subsamples, as in the cross-sectional case of Example~\ref{example:Heterogeneity-RCT}; dynamic treatment effects are
estimated from repeated observations of the same units and are correlated across horizons, while discretizing an outcome, as in Examples~\ref{example:LATE-instrument-validity} and~\ref{example:discrimination}, yields components that are cell
frequencies of a common multinomial, and hence correlated by construction.
\section{Notation and Setup}\label{sec:setup}
This section introduces the notation and formally presents the hypothesis of sign agreement, along with the test statistics. Additionally, section \ref{subsec:uniform-validity} explains the concept of the uniform asymptotic validity in contrast to the pointwise asymptotic validity, a common requirement for tests, and elucidates why pursuing the uniform asymptotic validity is desirable in our problem.
We begin with notation. Let \(\{W_i: i=1, \ldots, n\}\) be a random sample from $P$, and let \(\hat{P}_n\) denote the empirical distribution. Write $\mu(P)=(\mu_1(P),..., \mu_k(P))=\mathbb{E}_P[W_i]\). For each $j=1,...,k$, let \(\sigma_j^2(P)\) denote the variance of the $j$th component of $W_i$, and let \(\Omega(P)\) denote the correlation matrix of \(P\). Sample analogues are defined by replacing \(P\) with \(\hat{P}_n\). In particular, \(\bar{W}_n=(\bar{W}_{1,n},...,\bar{W}_{k,n})=\mu(\hat{P}_n)=(\mu_1(\hat{P}_{n}),...,\mu_k(\hat{P}_{n}))\), \(\hat{\Omega}_n=\Omega(\hat{P}_n)\), and \(S_{j, n}^2=\sigma_j^2(\hat{P}_n)\) for all \(j=1,..., k\); Define a diagonal matrix of sample variances as \(S_n^2=\operatorname{diag}(S_{1, n}^2, .., S_{k, n}^2)\). Let \(\mathbf{R}^k\) denote the \(k\)-dimensional Euclidean space, and define \(\mathbf{R}^k_+=\{x\in\mathbf{R}^k: x_j \geq 0 \text{ for all }j=1,...,k\}\) and \(\mathbf{R}^k_-=\{x\in\mathbf{R}^k: x_j \leq 0 \text{ for all }j=1,...,k\}\). Finally, let \(\mathbf O\) denote the set of \(k\times k\) positive definite correlation matrices, let \(I_k\) denote the $k\times k$ identity matrix, let \(0_k\) denote the \(k\)-dimensional zero vector, and let $[k]$ denote the set of index $\{1,...,k\}.$
We are interested in testing hypotheses:
\begin{equation}\label{def:null-hypo}
\begin{gathered}
H_0: P\in \mathbf{P}_0
\quad \text{versus}\quad
P\in \mathbf{P}\setminus\mathbf{P}_0, \\
\text{where}\quad
\mathbf{P}_0
=
\left\{
P\in\mathbf{P}:
\mathbb{E}_P[W_i]\leq 0_k
\quad\text{or}\quad
\mathbb{E}_P[W_i]\geq 0_k
\right\}
\end{gathered}
\end{equation}
where $\mathbf{P}$ is a large set of nonparametric distributions satisfying conditions in Theorems \ref{thm:LFtest-uniform-validity} and \ref{thm:conditional-uniform-validity} and the inequality applies elementwise. Throughout this article, $k$ is greater than or equal to 2, so $\mathbf P_0$ is nontrivial. For expositional clarity, the main text develops the tests for the sign-agreement hypothesis concerning the population mean vector. This formulation is not essential and extends to a broad range of regular estimands; see Appendix \ref{sec:extension-regular-estimators}.
The two proposed tests use the same test statistics. To introduce them, define three real-valued functions $T^{\ell}:\mathbf{R}^k \times\mathbf{O}\mapsto \mathbf{R}$, for $\ell=m,s,q$, by
\begin{align}\label{def:T-ell-function}
\begin{aligned}
T^{m}(x) &= \min\left\{\max_{1\leq j \leq k} \{x_j\}, \max_{1\leq j \leq k} \{-x_j\}\right\}
\\
T^s(x) &= \min\left\{\sum_{1\leq j \leq k} x_j^2 I\{x_j\geq0\}, \sum_{1\leq j \leq k} x_j^2 I\{x_j\leq0\}\right\}\\
T^q(x, \Omega) &= \inf_{\mu\in\mathbf R^k_+\cup \mathbf R^k_-} (x-\mu)' \Omega^{-1}(x-\mu)
\end{aligned}
\end{align}
where the second argument is suppressed for $\ell=m,s$. The superscripts $m$, $s$, and $q$ stand for maximum, sum of squares, and quasi-likelihood ratio (QLR), respectively.
All three functions share the property that $T^{\ell}(x)>0$ whenever $x\in\mathbf{R}^k$ contains components with opposite signs. They differ, however, in how they quantify the extent of this sign disagreement. The function \(T^{m}(x)\) is determined by the smaller of the largest positive component and the largest negative component in absolute value. The function \(T^{s}(x)\) is determined by the smaller of the sums of squared positive and negative components. Finally, \(T^{q}(x,\Omega)\) measures the squared Mahalanobis distance from \(x\) to the nearest vector whose components have a common sign. Interestingly, when $k=2$ and \(\Omega=I_2\), the three functions satisfy $T^s(x)=T^q(x,I_2)=\bigl(T^m(x)\bigr)^2$ whenever the components of $x$ have opposite signs. Accordingly, they generate the same family of level curves, as illustrated in Figure \ref{fig:Test-Statistic-Level-Curve}.
Using these functions, we define the test statistics by evaluating them at the vector of studentized sample means, $\sqrt{n}S^{-1}_n\bar{W}_n=(\tfrac{\sqrt{n}\bar{W}_{1,n}}{S_{1,n}},...,\tfrac{\sqrt{n}\bar{W}_{k,n}}{S_{k,n}})$, and the sample correlation $\hat{\Omega}_n$. For each $\ell\in\{m,s,q\}$, define
\begin{align}\label{def:test-statistic}
T^\ell_n = T^\ell(\sqrt{n}S^{-1}_n\bar{W}_n, \hat\Omega_n).
\end{align}
Large values of this test statistic provide evidence against the null hypothesis. These statistics encompass several test statistics used in the related literature. In particular, \cite{GailSimon1985Biometrics} and \cite{PiantadosiGail1993} use $T^s_n$ and $T^m_n$, respectively, for testing the sign agreement hypothesis. In the context of testing one-sided moment inequalities, \cite{Song2012JBES} employs $T^m_n$, the properties of which are further studied by \cite{Kim2023ET}.
\begin{figure}[tb]
\begin{center}
\begin{tikzpicture}[scale=0.90]
\fill[gray!40] (-3,-3) rectangle (0,0);
\fill[gray!40] (0,0) rectangle (3,3);
\draw[-latex] (-3,0) -- (3,0) node[right] {};
\draw[-latex] (0,-3) -- (0,3) node[above] {};
\draw[gray, thick, dashed] (1,-1) -- (3,-1) node[above] {$1$};
\draw[gray, thick, dashed] (1,-1) -- (1,-3) node[above] {};
\draw[gray, thick, dashed] (2,-2) -- (3,-2) node[above] {$2$};
\draw[gray, thick, dashed] (2,-2) -- (2,-3) node[above] {};
\draw[gray, thick, dashed] (-1,1) -- (-1,3) node[above] {};
\draw[gray, thick, dashed] (-1,1) -- (-3,1) node[above] {$1$};
\draw[gray, thick, dashed] (-2,2) -- (-3,2) node[above] {$2$};
\draw[gray, thick, dashed] (-2,2) -- (-2,3) node[above] {};
\node at (2.5, 0.5) {$H_0$};
\end{tikzpicture}
\end{center}
\caption{Level curves of $T^{\ell}(x)$ in \eqref{def:T-ell-function} for $x\in\mathbf{R}^2$. \footnotesize{The dashed curves represent the level curves of $T^{\ell}(x)$ in $\mathbf{R}^2.$ The two L-shaped curves marked by $1$ correspond to the set of values satisfying $T^m(x)=T^s(x)=T^q(x,I_2)=1$. Similarly, the two L-shaped curves marked by `2' correspond to the set of values satisfying $T^m(x)=2$ or $T^s(x)=T^q(x,I_2)=4$.}}
\label{fig:Test-Statistic-Level-Curve}
\end{figure}
\subsection{Uniform versus Pointwise Asymptotic Validity}\label{subsec:uniform-validity}
Consider a generic test $\phi_n=\phi(W_1,\ldots,W_n)$. Even when the null hypothesis is true, sampling randomness may lead the test to reject, resulting in a Type I error. Ideally, researchers wish to control the probability of such an error below a significance level $\alpha$ for every sample size $n$. This requirement is formalized as
\begin{align}\label{def:finite-sample-size}
\sup_{P\in \mathbf{P}_0} \mathbb{E}_P[\phi_n] \leq \alpha \quad \text{ for all }n\geq 1.
\end{align}
The left-hand side, which is the largest Type I error probability over the null, is called the \textit{finite-sample size} of the test. Because the requirement in \eqref{def:finite-sample-size} is difficult to attain beyond cases where the exact underlying distribution $P$ is known, researchers consider its asymptotic counterpart. A test $\phi_n$ is \textit{uniformly asymptotically valid} or \textit{uniformly asymptotically of level $\alpha$}, over $P\in \mathbf{P}_0$ if
\begin{align}\label{def:uniform-validity}
\limsup_{n\to\infty} \sup_{P\in\mathbf{P}_0}\mathbb{E}_P[\phi_n] \leq \alpha.
\end{align}
Uniform asymptotic validity ensures that the finite-sample size of the test is controlled less than or close to $\alpha$ in large samples.
Uniform asymptotic validity is stronger than the \textit{pointwise asymptotic validity} in the sense that it implies the latter. Specifically, a test $\phi_n$ is \textit{pointwise asymptotically valid} or \textit{pointwise asymptotically of level $\alpha$}, over $P\in \mathbf{P}_0$ if
\begin{align*}
\limsup_{n\to\infty} \mathbb{E}_P[\phi_n] \leq \alpha \text{ for all }P\in\mathbf{P}_0.
\end{align*}
The limit is taken for each distribution $P$, so the pointwise asymptotic validity does not guarantee the finite-sample size control even in large samples. For instance, there may exist a sequence of distributions $\{P_n\}^\infty_{n=1}\in\mathbf{P}_0$ such that $ \mathbb{E}_{P_n}[\phi_n] >\alpha$ for all $n\geq1$,
despite $\phi_n$ being pointwise asymptotically valid.
The distinction between pointwise asymptotic validity and uniform asymptotic validity is largely technical in many econometric applications: pointwise asymptotically valid tests often become uniformly asymptotically valid under reasonable assumptions on $\mathbf{P}$. However, this is not the case here. Tests that satisfy only the pointwise asymptotic validity may lead to size distortion under empirically relevant distributions, when the limiting distribution of the test statistic is discontinuous, as in our problem. Consequently, such test can be highly misleading even in large samples. For this reason, emphasis has been placed on uniform asymptotic validity in less-well behaved problems, including inference in partially identified models and one-sided moment inequalities tests; see, among others, \cite{ImbensManski2004confidence}, \cite{mikusheva2007uniform} and \textcolor{blue}{Andrews and Guggenberger} (\citeyear{AndrewsGuggenberger2009ECTA-hybrid}, \citeyear{AndrewsGuggenberger2010ET}), \textcolor{blue}{Romano and Shaikh (\citeyear{RomanoShaikh2008JSPI}, \citeyear{RomanoShaikh2010ECTA}}).
\section{Least Favorable Test}\label{sec:LFCtest}
This section proposes the least favorable (LF) test and presents one of the main results. Section \ref{subsec:LF-approach} describes a challenge in constructing such a test and the strategy used to circumvent it. This strategy underlies both the LF test and the conditional test proposed in Section \ref{sec:conditional-test}. Section \ref{subsec:LFC-critical-values} formally defines the critical value and the LF test. Section \ref{subsec:LF-result} presents uniform asymptotic validity of the LF test and related results.
\subsection{Least Favorable Approach}\label{subsec:LF-approach}
The construction of a uniformly asymptotically valid test based on the test statistic $T^{\ell}_n$ in \eqref{def:test-statistic} requires a critical value $\hat{c}^{\ell}_n$ satisfying
\begin{align}\label{def:section3-uniform-validity}
\limsup_{n\to\infty} \sup_{P\in\mathbf{P}_0} P\left\{T^{\ell}_n > \hat{c}^{\ell}_n\right\} \leq \alpha
\end{align}
for $\alpha\in(0,1)$. A seemingly natural choice for $\hat{c}^{\ell}_n$ is an estimator of the $1-\alpha$ quantile of the asymptotic distribution of $T^{\ell}_n$ under $P$. This choice, however, is infeasible because the distribution cannot be consistently estimated uniformly over $\mathbf{P}_0$.
To elucidate this challenge, we borrow the framework of \citet*{RomanoShaikhWolf2014ECTA} and \cite{CanayShaikh2017ARE}. For \(\theta_1\in\mathbf{R}^k\) and $P$, define the distribution function
\begin{align}\label{def:J_n-section3}
J^{\ell}_n(x, \theta_1,P)= P\left\{T^{\ell}(\sqrt{n}S^{-1}_{n} (\bar{W}_n-\mu(P)) + S^{-1}_n\theta_1)\leq x\right\}.
\end{align}
For fixed $\theta_1$, this distribution can be approximated by standard methods. In particular, if $\sqrt{n}S^{-1}_{n} (\bar{W}_n-\mu(P))$ is asymptotically normal, bootstrap methods can approximate this distribution, as discussed in Section \ref{subsec:LFC-critical-values}; continuity of function $T^{\ell}$ then ensures an approximation of $J^{\ell}_n(x, \theta_1,P)$.
The difficulty arises as the distribution of \(T_n^\ell\) corresponds to the nuisance parameter \(\theta_1=\sqrt{n}\mu(P)\):
\begin{align}\label{def:dist-test-stat}
J^{\ell}_n(x,\sqrt{n}\mu(P),P)
=P\left\{T^{\ell}(\sqrt{n}S^{-1}_{n} (\bar{W}_n-\mu(P)) + S^{-1}_n\sqrt{n}\mu(P))\leq x\right\}
=P\{T^{\ell}_n\leq x\}.
\end{align}
The nuisance parameter $\sqrt{n}\mu(P)$ not only varies with the sample size $n$, but is not consistently estimable uniformly over $\mathbf{P}_0$. For instance, when $\mu(P)=0_k$, its natural estimator $\sqrt{n}\bar{W}_n$ converges to a nondegenerate normal distribution, whereas the nuisance parameter itself equals $0_k$. This prevents uniform approximation of the distribution of $T^{\ell}_n$. The issue also appears in testing one-sided moment inequalities, as noted by \textcolor{blue}{Andrews and Guggenberger} (\citeyear{AndrewsGuggenberger2009ET}).
We circumvent this challenge adopting the least favorable (LF) approach. The idea is to use the largest quantile attainable over all values of $\sqrt{n}\mu(P)$ satisfying the null. Define
\begin{align}\label{def:c_n}
c^{\ell}_n(1-\alpha,P)=\sup\{(J^{\ell}_n)^{-1}(1-\alpha, \theta_1, P): \theta_1\in\mathbf{R}^k_+\cup\mathbf{R}^k_-\}
\end{align}
where $ (J^{\ell}_n)^{-1}(\xi,\theta_1, P)=\inf\{x\in\mathbf{R}:J^{\ell}_n(x,\theta_1,P) \geq \xi\}$ for any $\xi\in(0,1)$. By construction, $ c^{\ell}_n(1-\alpha,P)$ is no smaller than the $1-\alpha$ quantile of $T^{\ell}_n$ under the null, i.e.,
\begin{align*}
(J^{\ell}_n)^{-1}(1-\alpha, \sqrt{n}\mu(P), P) \leq c^{\ell}_n(1-\alpha,P) \quad\text{for all }\mu(P)\in \mathbf{R}^k_+\cup\mathbf{R}^k_-.
\end{align*}
Hence, the test that rejects when $T^{\ell}_n> c^{\ell}_n(1-\alpha,P)$ controls the finite-sample size:
\begin{align*}
\sup_{P\in\mathbf{P}_0} P\left\{T_n > c^{\ell}_n(1-\alpha,P)\right\} \leq \alpha \text{ at any }n\geq1.
\end{align*}
The next section formally constructs feasible critical values based on $ c^{\ell}_n(1-\alpha,P)$.
The LF approach to testing composite null hypotheses has a long history in statistics and econometrics. Seminal contributions include \cite{Perlman1969AoS} and the work of \textcolor{blue}{Wolak} (\citeyear{wolak1987JASA}, \citeyear{wolak1989ET}, \citeyear{wolak1991ECTA}), among many others. In the literature of testing sign agreement, this approach has been used in a finite sample setting; see, for example, \cite{GailSimon1985Biometrics}, \cite{PiantadosiGail1993}, \cite{RussekSimon1993Biometrics}, \cite{Silvapulle2001Biometrics}, and \citet*{MillerMolinariStoye2024WP}.
While the LF approach is also used in the literature on tests of one-sided moment inequalities, a notable distinction exists. The approach requires maximizing a quantile over the nuisance parameter. In our problem, the optimization in \eqref{def:c_n} does not have a unique solution, and its solutions vary with the distribution \(P\). By contrast, the corresponding optimization for one-sided moment inequalities has the unique constant solution \(0_k\). The root of this distinction lies in the non-monotonicity of the objective function in \eqref{def:c_n}. Specifically, whereas the largest quantile in one-sided moment inequality tests is always attained at the boundary point \(0_k\), the least favorable nuisance value in our problem depends on \(P\). This distinction makes testing sign agreement substantively different from testing one-sided moment inequalities.
\subsection{Least Favorable Critical Values} \label{subsec:LFC-critical-values}
Constructing estimators for $c^{\ell}_n(1-\alpha, P)$ in \eqref{def:c_n} hinges on the ability to consistently estimate the function $J^{\ell}_n(x, \theta_1, P)$ for fixed $\theta_1\in\mathbf{R}^k_+\cup\mathbf{R}^k_-$ and $x\in\mathbf{R}$. For convenience, we restate this distribution function:
\begin{align}\label{def:Jn-long}
J^{\ell}_n (x, \theta_1, P) =P\{T^{\ell}(\sqrt{n}S^{-1}_n (\bar{W}_n- \mu(P))+ S^{-1}_n \theta_1 ) \leq x\},
\end{align}
Although the centered studentized sample mean $\sqrt{n}S^{-1}_{n} (\bar{W}_n-\mu(P))$ depends on the unknown mean $\mu(P)$, its distribution is asymptotically normal and can therefore be estimated by parametric and nonparametric bootstrap methods.
Our critical value uses the parametric bootstrap. Define the function \(J^\ell\), indexed by \(\theta_1\in\mathbf R^k\) and \(\theta_2\in\bar{\mathbf O}\), by replacing the asymptotically normal component with the normal vector \(\theta_2^{1/2}Z\) where \(Z\sim N(0_k,I_k)\), and by reparameterizing \(S_n^{-1}\theta_1\) as \(\theta_1\):
\begin{align}\label{def:J}
J^\ell(x, \theta_1,\theta_2) = P\left\{T^\ell(\theta^{1/2}_2Z+ \theta_1, \theta_2) \leq x\right\}.
\end{align}
Here, the second argument of \(T^\ell\) is suppressed for \(\ell=m,s\). Then, $ J^{\ell}(x, \theta_1,\hat{\Omega}_n)$ provides a consistent estimator of $J^{\ell}_n (x, \theta_1, P)$. Analogously to \(c_n^\ell(1-\alpha,P)\) in \eqref{def:c_n}, define the critical value as the largest $1-\alpha$ quantile attainable over all $\theta_1$ satisfying the null:
\begin{align}\label{def:LF-cv-normal}
\begin{aligned}
{c}^{\ell}(1-\alpha, \hat\Omega_n)
&=\sup\left\{(J^{\ell})^{-1}(1-\alpha, \theta_1, \hat{\Omega}_n) : \theta_1\in\mathbf{R}^k_+ \cup\mathbf{R}^k_-\right\}\\
&=\sup\left\{(J^{\ell})^{-1}(1-\alpha, \theta_1, \hat{\Omega}_n) : \theta_1\in\mathbf{R}^k_+\right\}
\end{aligned}
\end{align}
where the second equality follows from symmetry. When $\hat\Omega_n = I_k$, this critical value simplifies to
\begin{align}
{c}^{\ell}(1-\alpha, I_k) &=\left\{
\begin{matrix}
\Phi^{-1}((1-\alpha)^{\frac{1}{k-1}}) & \quad \ell=m,\\
(1-\alpha)\text{-quantile of }\sum^{k-1}_{r=0}{k-1 \choose r}2^{-(k-1)}\chi^2_r & \quad \ell=s, q
\end{matrix}\right.\label{def:LF-cv-independent}
\end{align}
where the second line denotes a chi-bar-square mixture that assigns weight \({k-1\choose r}2^{-(k-1)}\) to a chi-square distribution with \(r\) degrees of freedom, and \(\chi_0^2\) is degenerate at zero.
As an alternative to \eqref{def:LF-cv-normal}, we also consider its nonparametric bootstrap counterpart. Let $\{W_1^*,\ldots,W_n^*\}$ be a sample drawn with replacement from the empirical distribution $\hat P_n$, and let $\bar W_n^*$ and $S_n^*$ denote the corresponding sample mean and diagonal matrix of sample standard deviations. Conditional on the data, the distribution of $T^\ell( \sqrt n\,S_n^{*-1}(\bar W_n^*-\bar W_n) +S_n^{*-1}\theta_1 )$ approximates $J_n^\ell(\cdot,\theta_1,P)$ for fixed $\theta_1$; see
\citet{RomanoShaikh2012AoS}. Accordingly, define the nonparametric bootstrap LF critical value by
\begin{gather}
\begin{aligned}
{c}^{\ell*}_n(1-\alpha,\hat{P}_n)
&= \sup\left\{(J^{\ell}_n)^{-1}(1-\alpha, \theta_1, \hat{P}_n): \theta_1\in\mathbf{R}^k_+ \cup\mathbf{R}^k_- \right\}.
\end{aligned}\label{def:LFC-bootstrap-critical-value}
\end{gather}
With the test statistic $T^\ell_n$ in \eqref{def:test-statistic} and two critical values $ c^{\ell}_n(1-\alpha, \hat{\Omega}_n)$ and $ c^{\ell*}_n(1-\alpha, \hat P_n)$ in \eqref{def:LF-cv-normal} and \eqref{def:LFC-bootstrap-critical-value}, we define the LF test by, for each $\ell=m,s,q,$
\begin{align}\label{def:LFtest}
\phi^{\ell}_n=I\left\{T^{\ell}_n> \hat{c}^{\ell}_n\right\} \quad\text{ where }\quad \hat{c}^{\ell}_n={c}^{\ell}(1-\alpha,\hat{\Omega}_n) \quad\text{or}\quad c^{\ell*}_n(1-\alpha,\hat{P}_n)
\end{align}
Thus, the researcher may choose among the three test statistics $\ell=m,s,q$ and between parametric and nonparametric bootstrap critical values.
Existing procedures developed specifically for sign agreement generally
impose stronger restrictions on the dependence structure. Most assume
$\Omega(P)=I_k$; see \cite{GailSimon1985Biometrics},
\cite{PiantadosiGail1993}, \cite{Silvapulle2001Biometrics}, and
\cite{LiChan2006detecting}. The test of \cite{RussekSimon1993Biometrics} permits a nonidentity correlation matrix \(\Omega(P)\) but is limited to \(k=2\) with known \(\Omega(P)\). \citet*{Zhao2019JASA} allow dependence only under an equicorrelated Gaussian structure. Hence, none of these procedures applies directly to the empirical setting described in Example \ref{example:Heterogeneity-RCT}.
Relatedly, our parametric-bootstrap critical values encompass existing critical values as special cases. When \(\hat\Omega_n=I_k\), the parametric critical values in \eqref{def:LF-cv-independent} coincide with that of \cite{PiantadosiGail1993} for \(\ell=m\) and those of \cite{RussekSimon1993Biometrics} for \(\ell=s,q\). When \(k=2\), our parametric critical value for \(\ell=s\) with a known correlation matrix coincides with that of \cite{RussekSimon1993Biometrics}, which was later rediscovered by \citet*{MillerMolinariStoye2024WP}. In contrast, our nonparametric-bootstrap critical values are novel. \cite{kowalski2023RES} proposes a heuristic nonparametric bootstrap method for the special case \(k=2\) and \(\Omega(P)=I_k\), but her approach does not extend beyond this setting, as shown by \citet*{MillerMolinariStoye2024WP}.
\begin{remark}
The critical values in \eqref{def:LF-cv-normal} are well defined, as established in Lemmas \ref{lem:parametric-cv-existence}--\ref{lem:argmax-critical-value-all} in the Appendix. In general, however, they do not admit tractable closed-form expressions and must typically be evaluated numerically, because the maximizing values of \(\theta_1\) depend on \(\hat\Omega_n\). For example, when \(k=2\) and \(\hat\Omega_n=I_2\), the supremum is attained in the limiting directions \((\infty,0)\) and \((0,\infty)\) for all $\ell=m,s,q$. When \(\hat\Omega_n=\left(\begin{smallmatrix}1&-1\\-1&1\end{smallmatrix}\right)\), the origin is a maximizer for $\ell=m,s$, and also for the natural continuous extension of $T^q$ to this singular correlation matrix.
\end{remark}
\begin{remark}\label{remark:regularized-correlation}
For \(\ell=q\), if \(\hat\Omega_n\) is not positive definite, replace it by $\tilde\Omega_n=
({\hat\Omega_n+\varepsilon_n I_k})/({1+\varepsilon_n}),$ where \(\varepsilon_n>0\) and \(\varepsilon_n\to0\), in both the test statistic \(T_n^q\) and the critical value \(\hat c_n^q\).
\end{remark}
\begin{remark}
The nonparametric bootstrap critical value in \eqref{def:LFC-bootstrap-critical-value} differs from that obtained by a naive application of the nonparametric bootstrap. Given the nonparametric bootstrap sample $\{W^*_1,...,W^*_n\}$, an analyst might instead compute a bootstrap analog of $T^{\ell}_n$ given as
\begin{align*}
T^{\ell*}_n=T^{\ell}\left(\left(\tfrac{\sqrt{n}\bar{W}^*_{1,n}}{S^*_{1,n}},...,\tfrac{\sqrt{n}\bar{W}^*_{k,n}}{S^*_{k,n}}\right),\hat\Omega^*_n\right)
\end{align*}
and use its $1-\alpha$ quantile of $T^{\ell*}_n$ as the critical value. In general, however, this procedure fails to consistently estimate the distribution of $T^{\ell}_n$. Such bootstrap failures for nonsmooth statistics involving maximum or minimum operators has been long recognized (e.g., \cite{bickel1981some}, \citet*{bickel1997resampling} and \cite{andrews2000inconsistency}). In the sign-agreement setting, \citet*{MillerMolinariStoye2024WP} further show for $k=2$ that related heuristic bootstrap procedures can have arbitrary rejection probabilities.
\end{remark}
\begin{remark}\label{rem:LF-IUT-comparison}
Consider $\Omega=I_k$ and the max statistic. A natural
intersection--union test separately tests $H_0^+:\mu\in\mathbf R_+^k$ and $H_0^-:\mu\in\mathbf R_-^k$ at level $\alpha$ and rejects sign agreement only when both component tests reject. Using the usual one-sided max tests, the critical value for each component is
\[
c_{\mathrm{IUT}}^m
=
\Phi^{-1}\left((1-\alpha)^{1/k}\right).
\]
Hence the resulting intersection--union test rejects when $T^m_n>c_{\mathrm{IUT}}^m.$ By contrast, the LF critical value in
\eqref{def:LF-cv-independent} is
\[
c_L^m
=
\Phi^{-1}\left((1-\alpha)^{1/(k-1)}\right)
<
\Phi^{-1}\left((1-\alpha)^{1/k}\right)
=
c_{\mathrm{IUT}}^m.
\]
Thus, although both procedures are valid ways to test sign agreement, the
direct LF construction exploits the union structure to use a strictly
smaller critical value. Its rejection region therefore contains that of
the corresponding intersection--union max test, so it is weakly more
powerful against every alternative, and strictly more powerful whenever
$T_n^m$ has positive probability between the two critical values.
\end{remark}
\subsection{Uniform Asymptotic Validity of the Least Favorable Test}\label{subsec:LF-result}
Our first main result establishes the uniform asymptotic validity of the LF test. This implies that, for sufficiently large $n$, the finite-sample size of the LF test is no greater than or close to the significance level $\alpha$.
\begin{theorem}\label{thm:LFtest-uniform-validity}
Let $W_i, i=1,..., n$, be an i.i.d. sequence of random vectors with distribution $P \in \mathbf{P}$ on $\mathbf{R}^k$. Suppose $\mathbf{P}$ satisfies the following uniform integrability condition:
\begin{align}\label{def:uniform-integrability}
\lim _{\lambda \rightarrow \infty} \sup _{P \in \mathbf{P}} \mathbb{E}_P\left[\left(\frac{W_{j, 1}-\mu_j(P)}{\sigma_j(P)}\right)^2 I\left\{\left|\frac{W_{j, 1}-\mu_j(P)}{\sigma_j(P)}\right|>\lambda\right\}\right]=0 \quad\text{for}\quad j=1,...,k.
\end{align}
Then, for $\alpha\in(0,\tfrac12)$ and $\ell\in\{m,s\}$, the least favorable test $\phi^{\ell}_n$ in \eqref{def:LFtest} satisfies the uniform asymptotic validity, i.e.,
\begin{align}\label{def:LFtest-uniform-validity}
\limsup_{n\to\infty} \sup_{P \in \mathbf{P}_0} \mathbb{E}_P [\phi^{\ell}_n] \leq \alpha .
\end{align}
Furthermore, if the smallest eigenvalue of $\Omega(P)$ is bounded away from zero uniformly over $\mathbf{P}$,
$\inf_{P \in \mathbf{P}} \lambda_{\min}(\Omega(P))>0$, then \eqref{def:LFtest-uniform-validity} holds for $\ell=q.$
\end{theorem}
The uniform integrability condition in \eqref{def:uniform-integrability} is both mild and standard. It is satisfied if any moment higher than the second exists uniformly over $\mathbf{P}$, i.e., for some $\delta>0$,
\begin{align*}
\sup _{P\in \mathbf{P}} \mathbb{E}_P\left[\left(\frac{W_{j,1}-\mu_j(P)}{\sigma_j( P)}\right)^{2+\delta}\right]<\infty \quad\text{for}\quad j=1,...,k.
\end{align*}
The condition is necessary to invoke a uniform central limit theorem and uniform law of large numbers, fundamental tools in facilitating uniform inference. As such, either this condition or a slightly stronger version stated above is frequently imposed in studies concerning uniform inference; see, among others, \cite{Romano2004SJS-uniformity}, \textcolor{blue}{Romano and Shaikh (\citeyear{RomanoShaikh2008JSPI}, \citeyear{RomanoShaikh2010ECTA}, and \citeyear{RomanoShaikh2012AoS})}, \cite{AndrewsGuggenberger2009ET}, and \cite{AndrewsSoares2010ECTA}.
Theorem \ref{thm:LFtest-uniform-validity} establishes that the asymptotic size of the LF test is no greater than $\alpha$ uniformly over $\mathbf P_0$. This upper bound need not be attained for an arbitrary class $\mathbf P$. To see why, recall that the distribution of the test statistic depends on the nuisance parameter in \eqref{def:dist-test-stat}, whereas the critical value in \eqref{def:LF-cv-normal} is constructed by taking the supremum over all admissible values of $\theta_1$, because this nuisance parameter is not uniformly estimable. If $\mathbf P$ excludes sequences of distributions for which the nuisance parameter approaches a value attaining this supremum, then the critical value may be asymptotically larger than the relevant quantile of the test statistic, and the asymptotic size may be strictly below $\alpha$. The bound is nevertheless sharp and hence the test is nonconservative whenever $\mathbf P$ is sufficiently rich to contain such sequences. The following corollary formally states that allowing location shifts, a mild richness condition, is sufficient for the bound to be sharp.
\begin{corollary}\label{cor:LFCtest-uniform-validity-equality}
Suppose the assumptions of Theorem \ref{thm:LFtest-uniform-validity} hold and that $\mathbf P$ contains all location shifts of some mean-zero distribution with identity correlation matrix and finite, strictly positive marginal variances. Then, for each $\ell=m,s,q$, the LF test in \eqref{def:LFtest} with critical values in \eqref{def:LF-cv-normal} satisfy \eqref{def:LFtest-uniform-validity} with equality.
\end{corollary}
To our knowledge, Theorem \ref{thm:LFtest-uniform-validity}, together with forthcoming Theorem \ref{thm:conditional-uniform-validity}, provides the first uniform asymptotic validity results for tests of sign agreement over a broad class of nonparametric distributions. Existing results instead establish finite-sample validity within restricted classes of normal distributions. In addition to normality, they typically impose either $\Omega(P)=I_k$ \citep{GailSimon1985Biometrics,PiantadosiGail1993,Silvapulle2001Biometrics}, an equicorrelated $\Omega(P)$ \citep*{Zhao2019JASA} or $k=2$ with a known correlation coefficient \citep{RussekSimon1993Biometrics}. These results do not directly extend to uniform asymptotic validity over a general class of nonparametric distributions. One complication is that the limiting distribution of the test statistics are not continuous, so standard arguments relying on continuity do not apply directly. To address this, we draw on the techniques developed in the literature on one-sided moment inequality tests, particularly \cite{AndrewsSoares2010ECTA}, \cite{AndrewsGuggenberger2009ET}, \citet*{RomanoShaikhWolf2014ECTA}.
Although Theorem \ref{thm:LFtest-uniform-validity} is stated for a vector of
sample means, this formulation is not essential. Appendix
\ref{sec:extension-regular-estimators} extends the LF test,
as well as the conditional test introduced in Section
\ref{sec:conditional-test}, to general regular estimators. In particular, the
same uniform validity conclusions hold when the estimator admits a uniformly
asymptotically linear representation with uniformly integrable standardized
influence functions and consistently estimated marginal scales and correlation
matrix; see Corollary \ref{cor:general-estimator-uniform-validity}. Under
standard regularity conditions, this framework includes commonly used estimands such as OLS, strongly identified IV and two-stage least-squares, and difference-in-differences coefficients.
Finally, a useful simplification is available for the max statistic under nonnegative dependence. In this case, the fixed identity-correlation critical value $c^m(1-\alpha,I_k)$ remains valid, eliminating the need for correlation-specific critical-value calculations.
\begin{corollary}\label{cor:LFCtest-nonnegative-cv}
Suppose the assumptions of Theorem
\ref{thm:LFtest-uniform-validity} hold. If all elements of $\Omega(P)$ is nonnegative, then a modified least favorable test with a fixed critical value in \eqref{def:LF-cv-normal} with $\ell=m$, $$\tilde\phi^{m}_n := I\{T^{m}_n > c^{m}(1-\alpha, I_k)\},$$
satisfies the uniform asymptotic validity over $\mathbf{P}_0.$
\end{corollary}
\section{Conditional Test} \label{sec:conditional-test}
The least favorable (LF) test achieves uniform asymptotic validity by guarding against all nuisance-parameter values consistent with the null hypothesis, but this protection may come at a cost in power because the test does not exploit information about the signs of the population means contained in the observed studentized sample means. The conditional test introduced in this section incorporates this information through a two-step conditional procedure, and aims to improve power against a broad range of alternatives while retaining uniform asymptotic validity. Section \ref{subsec:conditional-test} defines the conditional test and explains the rationale for its construction. Section \ref{subsec:conditional-test-result} presents our second main result, which establishes its uniform asymptotic validity.
\subsection{Two-step Procedure}\label{subsec:conditional-test}
The conditional test consists of two steps. Step I uses the studentized sample means to classify each coordinate as providing sufficiently strong evidence of a positive mean, sufficiently strong evidence of a negative mean, or insufficient evidence in either direction. Step II then employs a contingent testing rule determined by the screening outcome. If all coordinates are screened in the same direction, the test does not reject; if both positive and negative signs are screened, it rejects immediately. In the remaining cases, it applies either a one-sided test to the coordinates whose signs remain unresolved or, when no sign is screened, a conditional version of the LF test. The procedure is called the conditional test because, in these cases, the distribution used to construct the second-step critical value is conditional on the screening event observed in Step I. Accordingly, both the test statistic and its critical value may vary with the screening outcome. In this section, we formally introduce this procedure.
\paragraph{Step I: Screening Signs}
Fix a tuning parameter $\tau\in(0,\alpha)$. For a correlation matrix $\theta_2\in\bar{\mathbf O}$, define a screening threshold by
\begin{align}
\kappa_\tau(\theta_2)
=
\inf\left\{
x\in\mathbf R:
P\left\{
\max_{1\le j\le k}
(\theta_2^{1/2}Z)_j
\le x
\right\}
\ge 1-\tau
\right\},
\qquad
Z\sim N(0_k,I_k).
\label{def:kappa-screening-threshold}
\end{align}
When $\theta_2=I_k$, the threshold has the closed-form expression
\[
\kappa_\tau
=
\kappa_\tau(I_k)
=
\Phi^{-1}\left((1-\tau)^{1/k}\right).
\]
Using the estimated correlation $\hat\Omega_n$, Step I classifies each coordinate in $[k]=\{1,...,k\}$ according to
\begin{align}\label{def:conditioning-events}
\begin{aligned}
\mathcal I_+
&=
\left\{
j\in[k]:
\frac{\sqrt n\,\bar W_{j,n}}{S_{j,n}}
>
\kappa_\tau(\hat\Omega_n)
\right\},\\
\mathcal I_-
&=
\left\{
j\in[k]:
\frac{\sqrt n\,\bar W_{j,n}}{S_{j,n}}
<
-\kappa_\tau(\hat\Omega_n)
\right\},\\
\mathcal I_0
&=
\left\{
j\in[k]:
\left|
\frac{\sqrt n\,\bar W_{j,n}}{S_{j,n}}
\right|
\le
\kappa_\tau(\hat\Omega_n)
\right\}.
\end{aligned}
\end{align}
A coordinate belongs to $\mathcal I_+$ when its studentized sample mean exceeds the upper screening threshold, thereby providing sufficiently strong evidence of a positive population mean. Similarly, a coordinate belongs to $\mathcal I_-$ when it falls below the lower threshold. The remaining coordinates belong to $\mathcal I_0$; for these coordinates, Step I does not resolve the sign of the population mean. Thus, ``unresolved'' refers to the screening outcome rather than to the population mean being zero. The sets $\mathcal I_+$, $\mathcal I_-$, and $\mathcal I_0$ form a partition of $[k]$.
Step I summarizes the screening outcome through the following five indicator functions:
\begin{align}\label{def:conditional-cases}
\begin{aligned}
I_{\mathrm{null}}
&=
I\{\mathcal I_+=[k]\}
+
I\{\mathcal I_-=[k]\},\\
I_{\mathrm{alt}}
&=
I\{
\mathcal I_+\neq\varnothing,\
\mathcal I_-\neq\varnothing
\},\\
I_{\mathrm{neg}}
&=
I\{
\mathcal I_+=\varnothing,\
\mathcal I_-\neq\varnothing,\
\mathcal I_0\neq\varnothing
\},\\
I_{\mathrm{pos}}
&=
I\{
\mathcal I_+\neq\varnothing,\
\mathcal I_-=\varnothing,\
\mathcal I_0\neq\varnothing
\},\\
I_{\mathrm{origin}}
&=
I\{\mathcal I_0=[k]\}.
\end{aligned}
\end{align}
Precisely, $I_{\mathrm{null}}=1$ if all coordinates are screened in the same direction, either positive or negative; $I_{\mathrm{alt}}=1$ if at least one coordinate is screened as positive and another is screened as negative; $I_{\mathrm{neg}}=1$ if at least one coordinate is screened as negative while all remaining signs are unresolved; $I_{\mathrm{pos}}=1$ in the symmetric case; and $I_{\mathrm{origin}}=1$ if all signs are unresolved. By construction, the screening outcome must satisfy one of these indicators, i.e., $I_{\mathrm{null}}
+
I_{\mathrm{alt}}
+
I_{\mathrm{neg}}
+
I_{\mathrm{pos}}
+
I_{\mathrm{origin}}
=
1.$
The threshold in \eqref{def:kappa-screening-threshold} controls the probability of screening a coordinate in the direction opposite to the null. For example, when $\mu(P)\in\mathbf R_-^k$, the probability that at least one coordinate is screened as positive is asymptotically no greater than $\tau$. By symmetry, the same statement holds for negative screening when $\mu(P)\in\mathbf R_+^k$. This property allows the test to reject immediately when $I_{\mathrm{alt}}=1$ while reserving the remaining rejection probability $\alpha-\tau$ for Step II.
\paragraph{Step II: Conditional Subtests}
Step II operates conditional on the screening outcome in Step I. If all coordinates are screened in the same direction and $I_{\mathrm{null}}=1$, Step II does not reject the null hypothesis. If both positive and negative signs are screened and $I_{\mathrm{alt}}=1$, Step II rejects the null hypothesis. When one of $I_{\mathrm{neg}}$, $I_{\mathrm{pos}}$, or $I_{\mathrm{origin}}$ equals one, Step II uses a statistic and critical value tailored to the corresponding screening event.
First, suppose that $I_{\mathrm{neg}}=1$. In this case, at least one coordinate has been screened as negative, no coordinate has been screened as positive, and the signs of the coordinates in $\mathcal I_0$ remain unresolved. Conditional on the negative screening being correct, we rule out the nonnegative branch of the original null hypothesis in \eqref{def:null-hypo}. The relevant reduced null is therefore
\[
H^-_{0}:
\mu_j(P)\le0
\qquad
\text{for all }j\in\mathcal I_0.
\]
Evidence that any coordinate in $\mathcal I_0$ has a positive mean is sufficient to reject this reduced null. To measure the violation of this reduced null, define test statistics
\begin{align}
T_n^{\ell,-}
=
T^{\ell,-}
\left(
\sqrt n\,S_n^{-1}\bar W_n,~
\hat\Omega_n;~
\mathcal I_0
\right),
\qquad
\ell\in\{m,s,q\},
\label{def:negative-branch-test-statistic}
\end{align}
where, for any nonempty set $\mathcal I\subseteq[k]$,
\begin{align}\label{def:conditional-negative-functions}
\begin{aligned}
T^{m,-}(x;\mathcal I)
&=
\max_{j\in\mathcal I}x_j,\\
T^{s,-}(x;\mathcal I)
&=
\sum_{j\in\mathcal I}
x_j^2I\{x_j\ge0\},\\
T^{q,-}(x,\Omega;\mathcal I)
&=
\inf_{\mu\in\mathbf R_-^{|\mathcal I|}}
(x_{\mathcal I}-\mu)'
(\Omega_{\mathcal I\times\mathcal I})^{-1}
(x_{\mathcal I}-\mu).
\end{aligned}
\end{align}
Here, the second argument is suppressed for $\ell=m,s$. When $\Omega=I_k$, $T^{q,-}=T^{s,-}$ and so $T^{q,-_n}=T^{s,-}_n$.
To construct the critical value, let
$Z_\theta\sim N(\theta_1,\theta_2)$ and define the conditional distribution function
\begin{align}
L_{\mathcal I_0}^{\ell,-}
(t,\theta_1,\theta_2)
=
P\left\{
T^{\ell,-}
(Z_\theta,\theta_2;\mathcal I_0)
\le t
\ \middle|\
\begin{array}{l}
Z_{\theta,j}<-\kappa_\tau(\theta_2)
\quad\text{for all }j\in[k]\setminus\mathcal I_0,\\
|Z_{\theta,j}|\le\kappa_\tau(\theta_2)
\quad\text{for all }j\in\mathcal I_0
\end{array}
\right\}.
\label{def:Lminus-selective}
\end{align}
This is the distribution of the reduced statistic conditional on observing the screening event associated with $I_{\mathrm{neg}}=1$ and the particular unresolved set $\mathcal I_0$. Because the nuisance parameter cannot be uniformly consistently estimated for the similar reason discuss in Section \ref{subsec:LF-approach}, Step II follows the least favorable approach and searches $\theta_1$ over the reduced null parameter space $\mathbf R^k_-$. Define, for $\gamma\in(0,\tfrac12),$
\begin{align}
c_{\mathcal I_0}^{\ell,-}
(1-\gamma,\hat\Omega_n)
=
\sup\left\{
\left(
L_{\mathcal I_0}^{\ell,-}
\right)^{-1}
(1-\gamma,\theta_1,\hat\Omega_n):
\theta_1\in\mathbf R_-^k
\right\}.
\label{def:cv-conditional-neg-general}
\end{align}
When $\hat\Omega_n=I_k$, this critical value simplifies to
\begin{align}
c_{\mathcal I_0}^{\ell,-}
(1-\gamma,I_k)
=
\begin{cases}
-\Phi^{-1}\!\left[
\Phi(\kappa_\tau)
-
\{2\Phi(\kappa_\tau)-1\}
(1-\gamma)^{1/|\mathcal I_0|}
\right],
& \ell=m,\\[6pt]
\displaystyle
\inf\left\{
t:
\textstyle\sum_{r=0}^{|\mathcal I_0|}
{|\mathcal I_0|\choose r}
2^{-|\mathcal I_0|}
F_{\kappa_\tau}^{*r}(t)
\ge1-\gamma
\right\},
& \ell=s,q.
\end{cases}
\label{def:cv-conditional-neg}
\end{align}
Here, $F_{\kappa_\tau}^{*r}$ is the $r$-fold convolution of the CDF $F_{\kappa_\tau}(t)
= P\{\chi^2_1 \leq t \mid \chi^2_1 \leq \kappa_{\tau}^2\}=
\frac{2\Phi(\sqrt t)-1}
{2\Phi(\kappa_\tau)-1}$ where $\chi^2_1$ is a chi-squared random variable of degree 1, and
$F_{\kappa_\tau}^{*0}(t)=I\{t\ge0\}$. The conditional subtest used when $I_{\mathrm{neg}}=1$ is
\begin{align}
\psi_n^{\ell,-}(\gamma)
=
I\left\{
T_n^{\ell,-}
>
c_{\mathcal I_0}^{\ell,-}
(1-\gamma,\hat\Omega_n)
\right\}.
\label{def:conditional-test-neg}
\end{align}
Second, suppose that $I_{\mathrm{pos}}=1$. In this case, at least one coordinate has been screened as positive, no coordinate has been screened as negative, and the signs of the coordinates in $\mathcal I_0$ remain unresolved. The relevant reduced null is
\[
H^+_{0}:
\mu_j(P)\ge0
\qquad
\text{for all }j\in\mathcal I_0.
\]
By symmetry, evidence of a negative mean among the unresolved coordinates is sufficient for rejection. Step II therefore uses the sign-reversed statistic
\begin{align}
T_n^{\ell,+}
=
T^{\ell,-}
\left(
-\sqrt n\,S_n^{-1}\bar W_n,
\hat\Omega_n;
\mathcal I_0
\right),
\qquad
\ell\in\{m,s,q\}.
\label{def:positive-branch-test-statistic}
\end{align}
The corresponding critical value is
\begin{align}
c_{\mathcal I_0}^{\ell,+}
(1-\gamma,\hat\Omega_n)
=
c_{\mathcal I_0}^{\ell,-}
(1-\gamma,\hat\Omega_n),
\label{def:cv-conditional-pos}
\end{align}
and the conditional subtest used when $I_{\mathrm{pos}}=1$ is
\begin{align}
\psi_n^{\ell,+}(\gamma)
=
I\left\{
T_n^{\ell,+}
>
c_{\mathcal I_0}^{\ell,+}
(1-\gamma,\hat\Omega_n)
\right\}.
\label{def:conditional-test-pos}
\end{align}
Lastly, suppose that all coordinates fall within the screening region and $I_{\mathrm{origin}}=1$. Because Step I provides no directional information in this case, neither branch of the null hypothesis can be ruled out. Step II therefore retains the original statistic $T_n^\ell$ in \eqref{def:test-statistic} but constructs its critical value conditional on the event that all coordinates lie within the screening threshold. For $Z_\theta\sim N(\theta_1,\theta_2)$, define
\begin{align}
L^{\ell,0}
(t,\theta_1,\theta_2)
=
P\left\{
T^\ell(Z_\theta,\theta_2)\le t
\ \middle|\
|Z_{\theta,j}|
\le
\kappa_\tau(\theta_2)
\quad
\text{for all }j\in[k]
\right\}.
\label{def:Lzero-selective}
\end{align}
The least favorable conditional critical value is
\begin{align}
c^{\ell,0}
(1-\gamma,\hat\Omega_n)
=
\sup\left\{
(L^{\ell,0})^{-1}
(1-\gamma,\theta_1,\hat\Omega_n):
\theta_1\in\mathbf R_+^k
\right\}.
\label{def:cv-conditional-origin}
\end{align}
where we use that the supremum over $\mathbf R_+^k\cup\mathbf R_-^k$ is equal to the supremum over
$\mathbf R_+^k$ by sign symmetry. When $\hat\Omega_n=I_k$, it simplifies to
\begin{align}
c^{\ell,0}(1-\gamma,I_k)
=
\begin{cases}
-\Phi^{-1}\!\left[
\Phi(\kappa_\tau)
-
\{2\Phi(\kappa_\tau)-1\}
(1-\gamma)^{1/(k-1)}
\right],
& \ell=m,\\[6pt]
\textstyle
\inf\left\{
t:
\min_{a=0,\ldots,k}
F_{k,\kappa_\tau}^{(a)}(t)
\ge1-\gamma
\right\},
& \ell=s,q,
\end{cases}
\label{def:cv-conditional-origin-independence}
\end{align}
where, for $a=0,\ldots,k$,
\begin{align}
F_{k,\kappa_\tau}^{(a)}(t)
=
\textstyle\sum_{r=0}^{k-a}
{k-a\choose r}
2^{-(k-a)}
P\left\{
\min\left[
a\kappa_\tau^2
+
\textstyle\sum_{i=1}^{k-a-r}U_i',
\
\textstyle\sum_{i=1}^{r}U_i
\right]
\le t
\right\}.
\label{def:F-origin-conditional}
\end{align}
Here, $U_i$ and $U_i'$ are i.i.d. with CDF $F_{\kappa_\tau}$. The conditional subtest used when $I_{\mathrm{origin}}=1$ is
\begin{align}
\psi_n^{\ell,0}(\gamma)
=
I\left\{
T_n^\ell
>
c^{\ell,0}
(1-\gamma,\hat\Omega_n)
\right\}.
\label{def:phizero-selective}
\end{align}
\paragraph{Conditional Test}
Given a significance level $\alpha\in(0,\tfrac12)$ and a tuning parameter $\tau\in(0,\alpha)$, the conditional test is defined by
\begin{align}
\psi_n^\ell
=
I_{\mathrm{alt}}
+
I_{\mathrm{neg}}\,
\psi_n^{\ell,-}(\alpha-\tau)
+
I_{\mathrm{pos}}\,
\psi_n^{\ell,+}(\alpha-\tau)
+
I_{\mathrm{origin}}\,
\psi_n^{\ell,0}(\alpha-\tau),
\qquad
\ell\in\{m,s,q\}.
\label{def:conditional-test}
\end{align}
If $I_{\mathrm{null}}=1$, $\psi_n^\ell$ is set to zero and the test does not reject. If $I_{\mathrm{alt}}=1$, $\psi_n^\ell$ is set to one and the test rejects. If $I_{\mathrm{neg}}$, $I_{\mathrm{pos}}$, or $I_{\mathrm{origin}}$ equals one, the test is set equal to the corresponding conditional subtest at level $\alpha-\tau$.
One contribution of our paper lies in deriving the tractable representations of the conditional critical values in \eqref{def:cv-conditional-neg} and \eqref{def:cv-conditional-origin-independence}. The simplifications under $\hat\Omega_n=I_k$ follow from independence under the conditioning event. When $I_{\mathrm{neg}}=1$ or $I_{\mathrm{pos}}=1$, the coordinates in $\mathcal I_0$ are independent Gaussian variables truncated to $[-\kappa_\tau,\kappa_\tau]$, while the screened coordinates factor out of the conditional law. Stochastic monotonicity of the truncated Gaussian family, as established by \cite{lee2016AoS-post-selection-LASSO}, then implies that the least favorable conditional distribution under the reduced one-sided null is attained at the origin; see Proposition \ref{prop:conditional-lf-pos-branch} for details. Deriving the conditional critical value for the origin case $I_{\mathrm{origin}}=1$ is more involved. For $\ell=m$, the supremum in \eqref{def:cv-conditional-origin} is approached along mean vectors $\theta_1=(r,0,...,0)$ as $r\to\infty$. The key argument establishes Schur concavity first for $k=2$ and then extends it to arbitrary $k\geq2$ by induction; see Proposition \ref{prop:conditional-lf-origin-m} and its supporting lemmas. For $\ell=s,q$, a coordinatewise endpoint argument reduces the least favorable search to mean vectors whose coordinates are either zero or diverge in a common direction. If exactly $a$ coordinates diverge, their conditional distributions concentrate at the truncation boundary $\kappa_\tau$, while the remaining $k-a$ centered truncated coordinates generate the distribution $F_{k,\kappa_\tau}^{(a)}$. Taking the minimum over $a=0,\ldots,k$ yields the lower-envelope CDF in \eqref{def:cv-conditional-origin-independence}; see Proposition \ref{prop:conditional-lf-origin-sq} for details.
To the best of our knowledge, the conditional critical values in \eqref{def:cv-conditional-neg-general} and \eqref{def:cv-conditional-origin}, as well as their simplified forms in \eqref{def:cv-conditional-neg} and \eqref{def:cv-conditional-origin-independence}, are novel. The two-step approach in the presence of a nuisance parameter has been studied in statistics \citep{BergerBoos1994JASA,silvapulle1996JASA} and have also appeared in econometrics \citep*{StaigerStock1994ECTA,Chernozhukov2013intersection,McCloskey2017JoE,RomanoShaikhWolf2014ECTA}. Furthermore, the strategy of improving power against certain alternatives by incorporating information about the nuisance parameter has been extensively exploited in one-sided moment inequality tests; examples include \cite{Hansen2005JBES}, \cite{AndrewsSoares2010ECTA}, \cite{bugni2010ECTA}, \cite{Canay2010JoE}, \cite{andrewsbarwick2012ecta}, and \cite*{chernozhukov2019RES}, among others. Our conditional test draws particular inspiration from \citet*{RomanoShaikhWolf2014ECTA}, who construct a confidence set for the population mean in the first step and apply a least favorable procedure over that confidence set in the second step. Our procedure differs in three substantive respects. First, Step I classifies signs rather than constructing a confidence set. Second, both the statistic and its matching critical value depend on the realized screening outcome. Third, and most importantly, the second-step rejection probability is controlled conditional on that screening outcome.
\begin{remark}
For a general sample correlation $\hat\Omega_n$, the conditional critical values do not admit tractable closed-form expressions and must typically be evaluated numerically. When $\hat\Omega_n$ is not positive definite, the regularization convention in Remark \ref{remark:regularized-correlation} is applied in both the statistic and the conditional critical value for $\ell=q.$
\end{remark}
\subsection{Uniform Asymptotic Validity of the Conditional Test}\label{subsec:conditional-test-result}
Our second main result establishes the uniform asymptotic validity of the conditional test in \eqref{def:conditional-test}. Specifically, the maximal rejection probability over the null class is asymptotically bounded by the nominal significance level $\alpha$. Equivalently, for every $\varepsilon>0$, the worst-case null rejection probability is no greater than $\alpha+\varepsilon$ for all sufficiently large $n$.
\begin{theorem}\label{thm:conditional-uniform-validity}
Let $W_i, i=1, \ldots, n$, be an i.i.d. sequence of random vectors with distribution $P \in \mathbf{P}$ on $\mathbf{R}^k$. Suppose $\mathbf{P}$ satisfies the following uniform integrability condition in \eqref{def:uniform-integrability}. Then, for $\alpha\in(0,\tfrac12)$, $\tau\in(0,\alpha)$, and $\ell\in\{m,s\}$, the conditional test $\psi^{\ell}_n$ in \eqref{def:conditional-test} satisfies the uniform asymptotic validity, i.e.,
\begin{align}\label{def:conditional-test-uniform-validity}
\limsup_{n\to\infty} \sup_{P \in \mathbf{P}_0} \mathbb{E}_P [\psi^{\ell}_n] \leq \alpha.
\end{align}
Furthermore, if the smallest eigenvalue of $\Omega(P)$ is bounded away from zero uniformly over $\mathbf{P}$,
$\inf_{P \in \mathbf{P}} \lambda_{\min}(\Omega(P))>0$, then \eqref{def:conditional-test-uniform-validity} holds for $\ell=q.$
\end{theorem}
Theorem \ref{thm:conditional-uniform-validity} imposes the same distributional conditions as Theorem \ref{thm:LFtest-uniform-validity}. Theorems \ref{thm:LFtest-uniform-validity} and \ref{thm:conditional-uniform-validity} are the first results to establish uniform asymptotic validity for sign-agreement tests over such a broad class of nonparametric distributions. Sections \ref{subsec:LF-approach} and \ref{subsec:LF-result} discuss these assumptions in detail and compare them with those imposed by existing sign-agreement tests.
The validity argument reflects the specific structure of the conditional test. Each of the two steps of the conditional test may involve errors, and the parameter $\tau$ serves to balance the error rates between them. Explicitly, the tuning parameter $\tau$ controls the probability of a false directional screening. For example, under the nonnegative branch of the null, $\mu(P)\in\mathbf R_+^k$, any realization of $I_{\mathrm{alt}}=1$ or $I_{\mathrm{neg}}=1$ contains at least one negative screening. The probability of such an opposite-direction screening is asymptotically bounded by $\tau$. The screening-specific conditional subtests in Step II are calibrated at level $\alpha-\tau$. Because the relevant screening events are mutually exclusive, their combined contribution to the rejection probability is asymptotically bounded by $\alpha-\tau$. Thus, a Bonferroni-type decomposition gives the overall asymptotic size bound $\alpha$.
Researchers must choose the tuning parameter $\tau$ to implement the conditional test. Although every fixed choice $\tau\in(0,\alpha)$ satisfies Theorem \ref{thm:conditional-uniform-validity}, finite-sample power may depend on this choice. Increasing $\tau$ lowers the screening threshold $\kappa_\tau(\hat\Omega_n)$ and therefore makes Step I more likely to resolve signs, but it also reduces the level $\alpha-\tau$ allocated to the conditional subtests in Step II. Decreasing $\tau$ has the opposite effects: it makes directional screening more conservative while allocating more rejection probability to Step II. Determining an optimal choice of $\tau$ is beyond the scope of this article. In the simulation designs considered below, we recommend $\tau=\alpha/10$. The simulation results indicate that larger values of $\tau$ tend to reduce power, whereas reducing $\tau$ below $\alpha/10$ produces little additional improvement.
\section{Power Comparison}\label{sec:power-comparison}
In this section, we compare the rejection probabilities of our two proposed tests, the least favorable (LF) test and the conditional test. Because both tests are consistent against every fixed alternative, we focus on their rejection probabilities under local alternatives. For tractability, we restrict attention to the case in which the population correlation matrix is known and equal to $I_k$. We compare the LF test based on the parametric least favorable critical values in \eqref{def:LF-cv-independent} with the conditional test based on the critical values in \eqref{def:cv-conditional-neg} and \eqref{def:cv-conditional-origin-independence}.
For $A,B\geq0$ and integers $h,r\geq0$ satisfying $h+r\leq k$, consider the sequence of $P_n \in \mathbf{P}$ such that $\Omega(P_n) = I_k$ and
\begin{align}\label{eq:local-alternative}
\left(
\frac{\sqrt n\mu_1(P_n)}{\sigma_1(P_n)},\ldots,
\frac{\sqrt n\mu_k(P_n)}{\sigma_k(P_n)}
\right)
\to
\theta^{A,B,h,r}
=
(\underbrace{A,\ldots,A}_{h\text{ times}},
\underbrace{-B,\ldots,-B}_{r\text{ times}},
\underbrace{0,\ldots,0}_{k-h-r\text{ times}})
\end{align}
The most relevant alternatives have $h\geq1$ and $r\geq1$: $h$ coordinates exhibit positive local drift, $r$ coordinates exhibit negative local drift, and the remaining coordinates are locally centered.
For convenience, rewrite the conditional critical value \eqref{def:cv-conditional-neg} for the negative case subtest as
\begin{align}\label{eq:cv-conditional-onesided-rewritten}
\begin{aligned}
c_d^\ell
&=
c_{\mathcal I}^{\ell,+}(1-\alpha+\tau,I_k)
=
c_{\mathcal I}^{\ell,-}(1-\alpha+\tau,I_k).
\end{aligned}
\end{align}
where $\mathcal I\subset\{1,...,k\}$ with $|\mathcal I|=d$. The following proposition shows that neither procedure uniformly dominates the other. Instead, their relative performance depends on the dimension and on how many coordinates carry strong directional information.
\begin{proposition}\label{prop:power-comparison}
Suppose that the uniform integrability condition in \eqref{def:uniform-integrability} holds over $\mathbf{P}$. Let $P_n\in\mathbf{P}$ be a sequence of local alternatives satisfying \(\Omega(P_n)=I_k\) and \eqref{eq:local-alternative}. For $\alpha\in(0,\tfrac12)$ and $\tau\in(0,\alpha)$, consider the least favorable test \(\phi_n^\ell\) with $c^\ell(1-\alpha,I_k)$ in \eqref{def:LF-cv-independent} and the conditional test \(\psi_n^\ell\) with $c^{\ell}_d$ in \eqref{eq:cv-conditional-onesided-rewritten} and $c^{\ell,0}(1-\alpha+\tau, I_k)$ in \eqref{def:cv-conditional-origin-independence}.
\begin{enumerate}
\item[(i)] If \(k=2\), then, for each \(\ell\in\{m,s,q\}\), the least favorable test is asymptotically at least as powerful as the conditional test
for every fixed
\(\theta^{A,B,h,r}\in\mathbf R^2\):
\[
\lim_{n\to\infty}\mathbb E_{P_n}[\phi_n^\ell]
\geq
\lim_{n\to\infty}\mathbb E_{P_n}[\psi_n^\ell].
\]
\item[(ii)] Let \(k\ge3\) and \(h=1\). Then, for each \(\ell\in\{m,s,q\}\), there exists \(B_0^\ell>0\)
such that, for every \(r\in\{1,\ldots,k-1\}\) and every \(B\in(0,B_0^\ell]\), the least favorable test is asymptotically more powerful than the conditional test for sufficiently large \(A\):
\[
\lim_{n\to\infty}\mathbb E_{P_n}[\phi_n^\ell]
>
\lim_{n\to\infty}\mathbb E_{P_n}[\psi_n^\ell].
\]
\item[(iii)] Let \(k\ge3\) and \(h\ge2\). For \(\ell=m\), let
\(r\in\{1,\ldots,k-h\}\); for \(\ell\in\{s,q\}\), let \(r=k-h\).
Suppose that
\begin{align}\label{def:cv-comparison}
c_{k-h}^{\ell}<c^{\ell}(1-\alpha,I_k).
\end{align}
Then, for the corresponding \(\ell\), the conditional test is asymptotically more powerful than the least favorable test for sufficiently large \(A\):
\[
\lim_{n\to\infty}\mathbb E_{P_n}[\psi_n^\ell]
>
\lim_{n\to\infty}\mathbb E_{P_n}[\phi_n^\ell].
\]
\end{enumerate}
\end{proposition}
Proposition \ref{prop:power-comparison} highlights three distinct power regimes. First, when $k=2$, conditioning does not yield a local asymptotic power gain: the LF test weakly dominates the conditional test for every fixed local direction. Second, this lack of dominance is not confined to the two-dimensional case. When $k\geq3$ but only one coordinate has a strongly positive local drift, the LF test remains strictly more powerful when the negative drifts are sufficiently small. In this regime, screening out a single strongly positive coordinate does not provide enough dimensional reduction to compensate for the smaller second-step level $\alpha-\tau$ and the effect of conditioning.
Finally, the conditional test can become more powerful when at least two coordinates provide strong positive directional information. As $A\to\infty$, these $h$ coordinates are screened as positive with
probability approaching one, and the relevant conditional subtest is
effectively based on the remaining $k-h$ coordinates. Condition
\eqref{def:cv-comparison} requires the resulting $(k-h)$-dimensional
conditional critical value to be smaller than the LF critical value. This ordering is a convenient sufficient condition, but it is not necessary for the conditional test to be more powerful. In the proof, $c_{k-h}^{\ell}$ is used to construct a convenient subset of the conditional test's rejection event, so the resulting probability is only a lower bound on its limiting rejection probability. The conditional test may also reject under other screening configurations, including configurations in which fewer than $k-h$ coordinates remain unresolved and the applicable critical value is weakly smaller than $c_{k-h}^{\ell}$. Consequently, failure of \eqref{def:cv-comparison} does not imply that the LF test is more powerful.
Condition \eqref{def:cv-comparison} is straightforward to verify
numerically. For example, when $\alpha=0.05$ and $\tau=0.005$, it holds
for $\ell=m,s,q$ for
\[
(k,h)=(3,2),(4,2),(4,3),(5,2),(5,3),(5,4),
(6,2),(6,3),(6,4),(6,5).
\]
For the max statistic, this condition yields the desired power comparison
for every admissible $r$. For the sum of squares and quasi-likelihood ratio statistics, the critical-value ordering alone is sufficient when $r=k-h$.
\section{Simulation Study}\label{sec:MonteCarlo-Simulation}
This section examines the finite-sample performance of the least favorable (LF) test and the conditional test. We evaluate their maximum null rejection probabilities (MNRPs) and power at the significance level $\alpha=0.05$. Throughout, the sample size is $n=250$, the tuning parameter for the conditional test is $\tau=\alpha/10=0.005$, and the number of parameters is $k\in\{3,5,8\}$. For the power comparisons, we additionally report intersection--union adaptations of the one-sided moment-inequality procedures of \citet{Cox/Shi:2023} (CS) and \citet*{RomanoShaikhWolf2014ECTA} (RSW). Each procedure is applied in both directions, and sign agreement is rejected only when both one-sided null hypotheses are rejected. The RSW tuning parameter is set at $\beta=0.005$.
The simulation study uses normally distributed observations. For the MNRP exercise, observations are generated as $W_i\sim N_k(\mu,\Omega)$, $i=1,...,n$, and the tests are applied to the studentized sample mean $\sqrt n S_n^{-1}\bar W_n$ together with the sample correlation matrix $\hat\Omega_n$. We consider three correlation matrices, denoted by $\Omega_{\mathrm{Pos}}$, $\Omega_{\mathrm{Zero}}$, and $\Omega_{\mathrm{Neg}}$. The matrix $\Omega_{\mathrm{Zero}}$ is the identity matrix $I_k$. The other two matrices are Toeplitz matrices whose $(r,s)$ element is $\rho^{|r-s|}$, with $\rho=0.9$ for $\Omega_{\mathrm{Pos}}$ and $\rho=-0.9$ for $\Omega_{\mathrm{Neg}}$.
For general $\hat\Omega_n$, the LF critical values are calculated numerically. Since the nuisance-parameter space $\mathbf R_+^k$ is unbounded, we approximate it by the filled simplex
$\Theta_k
=
\{
\theta_1\in\mathbf R_+^k:
\textstyle\sum_{j=1}^k\theta_{1,j}\leq 10+k
\}.$
The candidate points are chosen to cover both concentrated and dispersed configurations of the nuisance mean. First, the origin is included. Second, along each coordinate axis, we consider $\theta_1=r e_j$ for
$r\in\{0.5,1,2,4,8,10+k\}$ and $j=1,\ldots,k$, where $e_j$ denotes the $j$th unit vector. These points allow the search to cover configurations ranging from small shifts near the origin to large shifts concentrated in a single coordinate. Third, additional points are generated from a Dirichlet distribution and rescaled to satisfy
$\sum_{j=1}^k\theta_{1,j}=10+k$. These points lie on the outer face of the simplex and allow the search to consider configurations in which the nuisance mean is distributed across several coordinates rather than concentrated along a coordinate axis. A first Monte Carlo step evaluates the critical value over all candidate points, after which the candidates producing the largest values are reevaluated using a larger Monte Carlo sample.
The general-correlation conditional critical values are computed in a similar two-step manner. For the origin case, the nuisance-parameter search uses the same type of simplex approximation. In the positive and negative cases, the unresolved coordinates are instead kept at the boundary value zero while increasingly large shifts are assigned to the screened coordinates. The Gaussian distribution is simulated conditional on the realized screening event using a weighted sequential-conditioning method. Thus, in the MNRP exercise, both the screening threshold and the applicable critical value are recomputed using the realized $\hat\Omega_n$ in each simulation repetition.
The empirical MNRPs are calculated over a collection of null mean vectors of the form
\begin{align}\label{eq:simulation-null-grid-style}
\mu_{j,M}
=
\frac{1}{\sqrt n}
(\underbrace{M,\ldots,M}_{j},0,\ldots,0)',
\qquad
j=0,\ldots,k,
\qquad
M\in\{5,10,15,20\}.
\end{align}
All of these mean vectors satisfy the nonnegative component of the null hypothesis. By sign symmetry, it is unnecessary to separately examine their nonpositive counterparts. For each correlation design and value of $k$, the empirical MNRP is obtained from the largest rejection frequency among the null configurations considered.
\begin{table}[htbp]
\centering
\caption{Empirical maximum null rejection probabilities (\%)}
\label{tab:mnrp-normal}
\begin{tabular}{lccccccccc}
\toprule
& \multicolumn{3}{c}{$k=3$}
& \multicolumn{3}{c}{$k=5$}
& \multicolumn{3}{c}{$k=8$} \\
\cmidrule(lr){2-4}
\cmidrule(lr){5-7}
\cmidrule(lr){8-10}
Test
& $\Omega_{\mathrm{Pos}}$
& $\Omega_{\mathrm{Zero}}$
& $\Omega_{\mathrm{Neg}}$
& $\Omega_{\mathrm{Pos}}$
& $\Omega_{\mathrm{Zero}}$
& $\Omega_{\mathrm{Neg}}$
& $\Omega_{\mathrm{Pos}}$
& $\Omega_{\mathrm{Zero}}$
& $\Omega_{\mathrm{Neg}}$ \\
\midrule
LF max
& 5.60 & 5.50 & 5.44
& 6.30 & 5.75 & 5.30
& 5.95 & 5.75 & 5.95 \\
LF sum
& 5.10 & 5.35 & 5.80
& 5.36 & 5.30 & 6.00
& 5.80 & 5.20 & 5.65 \\
LF QLR
& 5.70 & 5.50 & 5.68
& 6.25 & 5.24 & 5.35
& -- & -- & -- \\
\addlinespace
Conditional max
& 5.45 & 5.35 & 5.44
& 5.18 & 5.45 & 5.60
& 5.85 & 5.75 & 5.95 \\
Conditional sum
& 5.20 & 5.20 & 5.44
& 5.22 & 5.40 & 5.50
& 5.95 & 5.20 & 5.72 \\
Conditional QLR
& 5.35 & 5.25 & 5.54
& 5.05 & 5.34 & 6.00
& -- & -- & -- \\
\bottomrule
\end{tabular}
\end{table}
Table~\ref{tab:mnrp-normal} displays the empirical MNRPs of the six proposed tests. Overall, the rejection probabilities are close to the nominal significance level. Across the reported designs, they range from approximately $5.0\%$ to $6.3\%$. The largest rejection probability is observed around $6.3\%$ for $k=5$ under $\Omega_{\mathrm{Pos}}$, while the rejection probabilities for $k=8$ remain below approximately $6\%$ for all six procedures. The empirical MNRPs do not increase systematically with $k$, nor is the mild over-rejection concentrated in either the LF or the conditional test.
These results are broadly consistent with Theorems~\ref{thm:LFtest-uniform-validity} and~\ref{thm:conditional-uniform-validity}, which establish uniform asymptotic size control for the two tests. At the same time, the theorems do not imply exact finite-sample rejection probability of $5\%$, especially when the critical values are obtained numerically. Several features of the simulation can contribute to the small amount of over-rejection in Table~\ref{tab:mnrp-normal}: finite-sample studentization, estimation of the correlation matrix, Monte Carlo error in the critical values, and numerical approximation of the least favorable nuisance-parameter search. In addition, an empirical MNRP is itself a maximum over estimated rejection probabilities, so taking the maximum can produce a small upward Monte Carlo effect. Thus, the figure suggests that the finite-sample calibration is close to the nominal level, although it does not by itself identify the source of the remaining difference from $0.05$.
We next study empirical rejection probabilities under the alternative hypothesis. We restrict attention to $\Omega=I_k$ and $N(0,1)$ observations, and consider $k=3,5,$ and $8$. Each point in Figure~\ref{fig:simulation-power} is based on $10{,}000$ simulation repetitions with $n=250$. In addition to the LF and conditional tests based on the max and QLR statistics, we report the Cox--Shi test (CS QLR) and the two-step RSW procedure of based on the max and QLR statistics (RSW max and RSW QLR). For comparability, RSW is implemented using Gaussian parametric critical values rather than the nonparametric bootstrap. To relate the simulations to Proposition~\ref{prop:power-comparison}, note that the values of $A$ and $B$ reported in the figure are raw means, whereas the proposition is stated in terms of standardized local drifts. Thus, a raw mean $A$ corresponds approximately to $\sqrt n A$ in the proposition.
We consider three designs. Panel A sets $\mu=(A,-0.1,0,\ldots,0)',$ where only one coordinate has a positive mean. Panel B instead sets $\mu=(A,\ldots,A,-0.1)',$ so that $k-1$ coordinates have positive means. Finally, Panel C connects the two cases by gradually increasing the number of positive coordinates. For $h=0,\ldots,k-2$, the mean vector alternates between
\[
(\underbrace{A,\ldots,A}_{h},A/2,-B,0,\ldots,0)'
\quad\text{and}\quad
(\underbrace{A,\ldots,A}_{h+1},-B,0,\ldots,0)'.
\]
The horizontal axis in Panel C therefore represents the effective number of positive coordinates.
\begin{figure}[htbp]
\begin{center}
\caption{Empirical Alternative Rejection Probabilities}
\includegraphics[width=\textwidth]{power_all_designs.png}
\end{center}
\label{fig:simulation-power}
\footnotesize{Note: Panel A considers $\mu=(A,-0.1,0,\ldots,0)'$, Panel B considers $\mu=(A,\ldots,A,-0.1)'$, and Panel C gradually increases the number of positive coordinates with $A=0.4$ and $B=0.1$. All observations are generated from $N(\mu,I_k)$ with $n=250$, and each point is based on $10{,}000$ simulation repetitions. Blue and orange solid lines denote the LF and conditional tests, respectively. Green and purple dotted lines denote the CS and RSW procedures. Circle markers correspond to $T^m$-statistic procedures and square markers to $T^s$-statistic procedures.}
\end{figure}
Panel A shows that the LF and conditional tests have very similar power when only one coordinate provides strong positive directional information. Power increases with $A$ but decreases with $k$, since the same two nonzero means must be detected in the presence of more coordinates centered at zero. This pattern is consistent with the result in part (ii) of Proposition~\ref{prop:power-comparison}. Screening a single positive coordinate produces little reduction in the dimension of the remaining testing problem, and consequently the conditional test gains little relative to the LF test.
The pattern changes substantially in Panel B. When $k-1$ coordinates have positive means, the conditional test increasingly outperforms the LF test as $k$ grows. The difference is especially pronounced for the QLR statistic. For example, at $k=8$ and $A=0.4$, conditional QLR has power close to $47\%$, whereas LF QLR has power below $10\%$. This result corresponds directly to part~(iii) of Proposition~\ref{prop:power-comparison}. Here $h=k-1$ and $r=1=k-h$, so once the positive coordinates are screened, the conditional test effectively operates on a one-dimensional problem. The LF test, in contrast, continues to use a critical value that protects against nuisance configurations in the full $k$-dimensional problem. The resulting benefit from dimensional reduction becomes larger as $k$ increases.
Panel C makes this transition particularly transparent. When the effective number of positive coordinates is close to one, the LF and conditional tests have similar rejection probabilities, as in Panel A. As additional positive coordinates are introduced, the conditional tests gain power relative to the LF tests, and the difference becomes especially large for $k=5$ and $k=8$. For the max statistic, this pattern is directly consistent with part~(iii) of Proposition~\ref{prop:power-comparison}: at the integer points with at least two positive coordinates, $r=1\leq k-h$, and screening reduces the dimension of the second-step problem. For QLR, the proposition additionally requires $r=k-h$, which holds directly only at the endpoint $h=k-1$. The conditional-QLR gains at the intermediate points therefore extend beyond the sufficient condition established in the proposition. This is consistent with the discussion following Proposition~\ref{prop:power-comparison}: the critical-value ordering used there is sufficient, but not necessary, for the conditional test to be more powerful.
The comparison with the CS and RSW procedures reveals additional differences across the three designs. In Panel A, where only one coordinate has a positive mean, both the CS and RSW procedures generally have lower power than the proposed tests. The difference is particularly visible for larger $k$. Thus, when the alternative is relatively sparse, neither the active-constraint adjustment of CS nor the moment-selection adjustment of RSW provides an appreciable advantage over the LF and conditional tests.
The ranking changes in Panels B and C, where the alternative becomes increasingly dense. In Panel B, the CS test can have higher power at relatively small values of $A$, especially for $k=5$ and $k=8$. Its power then levels off, however, whereas the conditional tests continue to gain power as $A$ becomes larger and eventually overtake CS. This pattern is intuitive given the different forms of adaptation. CS can respond strongly to moderate evidence accumulated across several coordinates. The conditional test instead gains most when Step I can classify several coordinates as clearly positive, after which the second-step problem is substantially lower dimensional.
The RSW procedures display a pattern closer to that of the conditional tests. Their moment-selection step also exploits information about inequalities that appear sufficiently far from the boundary, and their power generally rises as more positive coordinates become separated from zero. In Panel B, RSW is typically below the conditional tests for intermediate values of $A$. Panel C shows the same pattern more continuously. As the effective number of positive coordinates increases, both the conditional and RSW procedures gain relative to the LF tests, particularly for $k=5$ and $k=8$.
Overall, the comparison indicates that the proposed conditional test is particularly competitive in the region emphasized by the theoretical power analysis: alternatives in which several coordinates provide sufficiently strong directional information to reduce the effective dimension of the problem. CS can be advantageous when several coordinates contain moderate rather than strong evidence, while RSW provides a closer competitor because its moment-selection mechanism also adapts to slackness. Across the designs considered here, however, no test uniformly dominates the others.
\section{Empirical Illustration}\label{sec:Empirical-Application}
This section applies the two proposed tests in empirical settings where
the existing sign agreement tests do not apply. In the first, we test
the validity of the college-proximity instrument of \citet{card1993NBER}.
In the second, we test whether the dynamic response of divorce rates to
unilateral divorce laws \citep{Wolfers2006AER} maintains a single sign
over event time. In both settings the components of $\hat\mu$ are
correlated by construction---cell frequencies of a common multinomial in
the first, coefficients of a single event-study regression in the
second---so the correlation matrix $\Omega$ is far from the identity.
\subsection{Validity of the College-proximity Instrument}
\citet{card1993NBER} studies the return to college in the United States using
the instrumental variable strategy of Example~\ref{example:LATE-instrument-validity} in Section \ref{sec:examples}: a binary instrument, a binary treatment, and a TSLS estimand interpreted as a local average treatment effect. The instrument is an indicator for having grown up near a four-year college. A local college lowers the cost of attendance, importantly by allowing students to enroll while living at home, and the causal interpretation of the estimand requires that this cost shift move everyone it moves in a single direction. Proximity may encourage degree completion: students from low-income families who would otherwise stop at high school can afford a degree when college is local, so $D_1\geq D_0$ and defiers are absent. On the other hand, proximity may increase enrollment without increasing degree completion: students whom a local college draws into higher education may rarely continue to a degree, while some students diverted from better colleges elsewhere may stop short of a degree they would otherwise have completed (see \citealt{rouse1995JBES}; \citealt{CohodesGoodman2014AEJ}). If proximity moves students across the degree-completion threshold only in this latter direction, then $D_1\leq D_0$ and compliers are absent.
Which force operates is an empirical question, and when the two operate on different students, neither direction of monotonicity holds and the TSLS estimand aggregates offsetting complier
and defier effects.
We test the null hypothesis in \eqref{eq:instrument-validity-null} using the standard extract of the National Longitudinal Survey of Young Men: an i.i.d.\ sample of $3{,}010$ men with valid 1976 wages and schooling, of
whom $2{,}053$ grew up near a four-year college ($Z=1$) and $957$ did not
($Z=0$). Following \citet{Kitagawa2015ECTA}, the treatment is degree
completion, $D=1\{\text{years of schooling}\geq 16\}$. The outcome is
the log hourly wage in 1976, discretized at its pooled-sample quartiles
($5.98$, $6.29$, and $6.56$ log points), so that $k=4$ in the notation of Example~\ref{example:LATE-instrument-validity} and the moment vector $\mu=(\mu_1,\dots,\mu_8)$ of \eqref{eq:instrument-validity-parameters} is eight-dimensional. We report both the max statistic $T^m$ and sum-of-squares statistic $T^s$, under the least-favorable and conditional ($\tau=0.1\alpha$) procedures, as the multinomial covariance matrix is not full rank.
Table~\ref{tab:card-results} reports the results. In Panel~A the estimated moments split cleanly by treatment
block: all four degree-holder moments are positive, the top wage bin strongly so ($z=4.64$), while among men without degrees the two top wage bins are significantly negative ($z=-2.57$ and $-4.05$), so that high-wage non-degree men are substantially more prevalent among those who grew up near a college. The least-favorable procedure yields $p$-values of $0.03\%$ and $0.04\%$ across the two statistics and the conditional procedure yields $p$-values below $0.1\%$, so sign agreement is rejected at the 1\% significance level; the rejection survives coarsening to terciles ($0.002\%$ and $0.02\%$ under the least-favorable procedure; Panel~B). This sharpens the rejection reported by \citet{Kitagawa2015ECTA}: his null maintains the no-defier direction $D_1\ge D_0$, whereas \eqref{eq:instrument-validity-null} is implied by the instrument validity under either direction of monotonicity, so its rejection refutes the model without taking a stand on which subpopulation is absent. Accordingly, the TSLS estimand in this application identifies neither the
average treatment effect for compliers nor that for defiers.
\begin{table}[t]\label{tab:card-results}
\centering
\caption{Sign agreement tests, \cite{card1993NBER} college-proximity instrument.}
\begin{tabular}{lcccc}
\toprule
\multicolumn{5}{l}{\emph{Panel A: Headline specification, $D=1\{\text{years}\geq16\}$, quartile bins $(2k=8)$}}\\
Wage bin & \multicolumn{2}{c}{$D=1$ block} & \multicolumn{2}{c}{$D=0$ block} \\
& $\hat\mu_j$ (SE) & $z$ & $\hat\mu_{4+j}$ (SE) & $z$ \\
\midrule
$(-\infty, 5.98]$ & $0.0027$ (0.0076) & $0.36$ & $\phantom{-}0.1308$ (0.0169) & $\phantom{-}7.72$ \\
$(5.98, 6.29]$ & $0.0139$ (0.0086) & $1.63$ & $\phantom{-}0.0276$ (0.0158) & $\phantom{-}1.75$ \\
$(6.29, 6.56]$ & $0.0011$ (0.0100) & $0.11$ & $-0.0373$ (0.0145) & $-2.57$ \\
$(6.56, \infty)$ & $0.0508$ (0.0109) & $4.64$ & $-0.0525$ (0.0130) & $-4.05$ \\
\midrule
\multicolumn{5}{l}{\emph{Panel B: Joint $p$-values (\%)}}\\
Specification & LF $T^m$ & LF $T^s$ & Cond $T^m$ & Cond $T^s$ \\
\midrule
Quartiles ($2k=8$) & 0.028 & 0.044 & 0.045 & 0.023 \\
Terciles ($2k=6$) & 0.002 & 0.018 & $\leq 0.001$ & 0.012 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.92\textwidth}
\vspace{2pt}
{\footnotesize\emph{Notes:} $n=3{,}010$ ($n_1=2{,}053$, $n_0=957$),
treated as i.i.d.\ with no weights. Wage-bin edges are pooled-sample
quantiles of log hourly wage; intervals are
right-closed. Both specifications use $D=1\{\text{years}\geq 16\}$. Sign agreement $p$-values, in percent, are reported for the least-favorable (LF) and conditional (Cond) procedures across the three statistics $(m,s)$.
}
\end{minipage}
\end{table}
\subsection{Dynamic Treatment Effects of Unilateral Divorce Law}
\citet{Wolfers2006AER} studies the effect of unilateral divorce laws on the divorce rate in the United States. Under a consent regime a divorce requires the agreement of both spouses, whereas under a unilateral regime either spouse may exit the marriage alone. Between 1968 and 1988 twenty-nine states switched from some variant of consent divorce to a unilateral system. Exploiting this staggered adoption, the paper estimates the dynamic impact of the policy on divorce rate using an unbalanced panel of state divorce rates for 1956--1988 ($n=1,631$). Let
\[
\mu_k \;=\; k\text{th year-bin dynamic treatment effect},
\qquad k = 1,\dots,8,
\]
denote the parameter of interest: the effect on the divorce rate (annual divorces per $1{,}000$ persons) of unilateral divorce having been in effect for the $k$th two-year bin, where $k=1$ is the first two years, $k=7$ is years 13--14, and $k=8$ collects year 15 onward. The corresponding estimates are reported in Table 2 of \citet{Wolfers2006AER} under three different specifications.
We test the null hypothesis in \eqref{def:intro-null} with $\mu = (\mu_1,\dots,\mu_8)$. Rejection is evidence that the dynamic response genuinely crosses zero, and this distinction carries the economics. Unilateral divorce lowers the cost of exiting a marriage: either spouse may now leave alone, and a thicker remarriage market and reduced stigma lower that cost further, so more marriages dissolve and $\mu_k \ge 0$. On the other hand, couples may match more cautiously in anticipation of easier dissolution, so marriages formed after the reform may be better selected and \(\mu_k \le 0\). Which force dominates is an empirical question, and the short-run and long-run effects may differ. Sign agreement is the formal boundary between a response of a single sign and one that reverses over event time. It is also a precondition for the conventional scalar summary. When \eqref{def:intro-null} holds, the single pooled treatment coefficient is a weighted average of like-signed quantities and inherits their sign, whereas when it fails the same coefficient aggregates offsetting terms and may sit near zero even though no individual $\mu_k$ is small.
\begin{figure}[htbp]\label{fig:wolfers}
\centering
\caption{Replicated estimates of $\mu_1,\dots,\mu_8$ from Table~2 of
\citet{Wolfers2006AER}}
\includegraphics[width=0.8\textwidth]{table_2_columns_1_2_two_panel_uniform_ci_90_95_99.png}
\begin{minipage}{0.92\textwidth}
\vspace{2pt}
{\footnotesize\emph{Notes:} The top panel corresponds to column~1 (state and year fixed effects) and the bottom panel corresponds to column~2 (adding state-specific linear trends). Shaded regions are $90\%$, $95\%$ and $99\%$ uniform confidence bands; vertical segments are $95\%$ pointwise intervals. Sign agreement $p$-values, in percent, are reported for the least-favorable (LF) and conditional (Cond) procedures across the three statistics $(m,s,q)$.}
\end{minipage}
\end{figure}
Figure~\ref{fig:wolfers} displays replicated estimates of $\mu_1,...,\mu_8$ together with our sign agreement test results, for the specification with state and year fixed effects (column 1, top panel) and the specification that adds state-specific linear time trends (column 2, bottom panel) in Table 2 of \cite{Wolfers2006AER}.\footnote{We omit the third specification (column 3 in Table 2 of \cite{Wolfers2006AER}), which adds state-specific quadratic time trends, because all point estimates are non-negative and the sign agreement p-value is trivially one.} In the top panel, least-favorable procedure yields $p$-values of $0.6$, $0.4$ and $0.3$ percent across the three statistics and the conditional procedure yields $0.6$, $0.5$ and $0.6$ percent, so sign agreement is rejected at the 1\% significance level; in the bottom panel the least-favorable procedure yields $11$, $14$ and $12$ percent while the conditional procedure yields $6.6$, $7.7$ and $6.6$ percent, a rejection at the 10\% level under conditional test alone. The evidence that the divorce-rate response reverses sign is thus strong under the first specification, suggestive under the second, implying that the long-run effect of unilateral divorce could differ from its short-run effect.
Methodologically, the conditional test yields smaller $p$-values than the least favorable in the presence of large and individually significant point estimates, consistent with the theoretical result in Proposition \ref{prop:power-comparison}. Finally, the uniform confidence band does not reveal the sign reversal at the 99\% level in the top panel, nor even at the 90\% level in the bottom panel, illustrating that our method is useful when analysis seeks to test sign agreement.
\section{Conclusion}\label{sec:Conclusion}
This article develops least favorable and conditional tests of sign agreement among a finite collection of parameters. The least favorable test uses the largest critical value over all null-consistent nuisance configurations, whereas the conditional test first screens coordinates with strong directional evidence and then tests the remaining sign-unresolved coordinates conditional on the screening outcome. Both procedures accommodate arbitrary correlation among the estimators and can be implemented using the maximum, sum-of-squares, or quasi-likelihood ratio statistic. We establish their uniform asymptotic validity and consistency against fixed alternatives. Although the main analysis is presented for studentized sample means, the results also apply to regular asymptotically linear estimators and smooth functions of moments under the stated conditions.
The theoretical and simulation results reveal a systematic power tradeoff. The LF test is competitive when directional information is sparse and screening produces little dimensional reduction. The conditional test can instead achieve substantial power gains when several coordinates are well separated from zero, because its second step operates on a smaller set of sign-unresolved coordinates. As a practical guideline, we recommend the LF test for $k=2$ and the conditional test as a default for $k\geq3$, while noting that the LF test can be preferable when directional information is sparse. The empirical maximum null rejection probabilities remain close to the nominal level across the dimensions and correlation structures considered. Comparisons with intersection--union adaptations of moment-inequality procedures show that no procedure considered dominates uniformly. The empirical illustrations demonstrate the usefulness of the proposed tests: they reject the testable implication of instrument validity in the college-proximity application and provide evidence of a sign reversal in the dynamic effect of unilateral-divorce laws.
The screening-and-conditioning principle underlying the conditional test is not specific to sign agreement and may also be useful for one-sided moment-inequality testing, where clearly slack moments could be screened out before testing the remaining, potentially binding inequalities conditional on the screening event. Developing the theory for such procedures is left for future research. Other extensions include allowing the number of parameters to increase with the sample size, or extending the framework to an infinite collection of parameters, which would permit applications such as testing sign agreement over a continuum of treatment-effect horizons or testing instrument validity with a continuous outcome without discretization. In addition, while rejection of sign agreement establishes that at least one parameter is positive and another is negative, it does not identify their signs while controlling simultaneous directional errors, suggesting the value of a post-rejection sign-classification procedure. Finally, implementation under a general correlation matrix requires numerical evaluation of least favorable and conditional critical values; further analytical characterizations of least favorable configurations, more accurate and computationally efficient numerical methods, and data-dependent choices of the screening level $\tau$ could improve practical implementation.
\phantomsection
\bibliography{draft_2026/unif2sided}
\addcontentsline{toc}{section}{References}
\appendices
\setcounter{equation}{0}
\small
\addcontentsline{apc}{appendices}{\protect\numberline{\thesubsection}Title of Appendix A}
\section{Extension to Regular Estimators}\label{sec:extension-regular-estimators}
The sample-mean formulation used in the main text is convenient but is not
essential for the proposed tests. This section shows that the least favorable (LF) test with
the parametric bootstrap critical value and the conditional test remain
uniformly asymptotically valid for a general regular estimator. We also briefly
discuss familiar estimators covered by the high-level conditions below.
Throughout, $k\ge2$ is fixed, as in the main text.
Let $U_i$, $i=1,\ldots,n$, be an i.i.d. sequence with distribution
$P\in\mathbf P$, and let $\beta(P)=(\beta_1(P),\ldots,\beta_k(P))'\in
\mathbf R^k$ be the parameter of interest. The null hypothesis is
\begin{align}\label{def:null-general-estimator}
H_0:\quad \beta(P)\in\mathbf R_+^k\cup\mathbf R_-^k,
\end{align}
and write
\[
\mathbf P_{\beta,0}
:=
\{P\in\mathbf P:\beta(P)\in\mathbf R_+^k\cup\mathbf R_-^k\}.
\]
Suppose that $\hat\beta_n$ estimates $\beta(P)$ and admits an influence
function $\zeta_P:\mathcal X\to\mathbf R^k$ satisfying
$\mathbb E_P[\zeta_P(U_i)]=0_k$. Define
\begin{align*}
V(P)&:=\mathbb E_P[\zeta_P(U_i)\zeta_P(U_i)'],\\
\sigma_j^2(P)&:=V_{jj}(P),\qquad
D(P):=\operatorname{diag}(\sigma_1(P),\ldots,\sigma_k(P)),\\
\Omega(P)&:=D(P)^{-1}V(P)D(P)^{-1},
\end{align*}
where $0<\sigma_j^2(P)<\infty$ for every $P\in\mathbf P$ and
$j\in[k]$. Let $\hat V_n$ be a symmetric positive-semidefinite estimator of
$V(P)$. On the event that all diagonal elements of $\hat V_n$ are strictly
positive, let $\hat D_n$ be the diagonal matrix formed from their square roots
and define
\[
\hat\Omega_n:=\hat D_n^{-1}\hat V_n\hat D_n^{-1}.
\]
On the complement of this event, set $\hat D_n=I_k$ and
$\hat\Omega_n=I_k$, and define the test using these values. The consistency
condition below implies that this convention is asymptotically irrelevant. If
a covariance estimator is not positive semidefinite by construction, it may
instead be replaced by a consistent positive-semidefinite projection before
forming $\hat D_n$ and $\hat\Omega_n$.
For $\ell\in\{m,s,q\}$, define
\begin{align}\label{def:general-estimator-statistic}
T_{n,\beta}^\ell
:=T^\ell\left(
\sqrt n\,\hat D_n^{-1}\hat\beta_n,
\hat\Omega_n
\right),
\end{align}
where the second argument is suppressed for $\ell=m,s$. The extension of
the LF test considered in this section uses the Gaussian parametric critical
value:
\begin{align}\label{def:general-estimator-parametric-LF-test}
\phi_{n,\beta}^{\ell}
:=I\left\{
T_{n,\beta}^\ell>c^\ell(1-\alpha,\hat\Omega_n)
\right\}.
\end{align}
Let $\psi_{n,\beta}^\ell$ denote the conditional test obtained from
$\psi_n^\ell$ by replacing $\sqrt n S_n^{-1}\bar W_n$ with
$\sqrt n\,\hat D_n^{-1}\hat\beta_n$ and using $\hat\Omega_n$ in the
screening rule, test statistics, and conditional critical values. Throughout,
the negative-case critical value is understood to take its supremum over
$\mathbf R_-^k$, and the positive-case critical value over $\mathbf R_+^k$,
as dictated by the corresponding reduced null hypotheses. For $\ell=q$, the
same regularization convention for a non-positive-definite $\hat\Omega_n$ as
in the main text is maintained.
\begin{assumption}
\label{ass:regular-asymptotically-linear}
The following statements hold for every sequence $\{P_n\}\subset\mathbf P$.
\begin{enumerate}[(i)]
\item The estimator is uniformly asymptotically linear after
studentization:
\begin{align}\label{eq:uniform-asymptotic-linearity-general}
\sqrt n\,D(P_n)^{-1}
\bigl(\hat\beta_n-\beta(P_n)\bigr)
=
\frac{1}{\sqrt n}\sum_{i=1}^n
\varphi_{P_n}(U_i)
+o_{P_n}(1),
\end{align}
where
\[
\varphi_P(U_i):=D(P)^{-1}\zeta_P(U_i).
\]
\item For each $j\in[k]$, the squared normalized influence functions are
uniformly integrable:
\begin{align}\label{eq:UI-normalized-influence-function}
\lim_{\lambda\to\infty}
\sup_{P\in\mathbf P}
\mathbb E_P\left[
\varphi_{P,j}(U_i)^2
I\{|\varphi_{P,j}(U_i)|>\lambda\}
\right]
=0.
\end{align}
\item The scale and correlation estimators are well defined with
probability approaching one and are uniformly consistent in the
sequential sense:
\begin{align}\label{eq:studentization-general-well-defined}
P_n\!\left\{\min_{j\in[k]}(\hat V_n)_{jj}>0\right\}\to1,
\end{align}
and
\begin{align}\label{eq:studentization-general-consistency}
\hat D_n^{-1}D(P_n)\overset{P_n}{\to}I_k,
\qquad
\|\hat\Omega_n-\Omega(P_n)\|_{\max}
\overset{P_n}{\to}0.
\end{align}
\end{enumerate}
\end{assumption}
Condition \eqref{eq:UI-normalized-influence-function} is the influence-function analogue of the standardized uniform integrability condition in \ref{def:uniform-integrability}.
Assumption \ref{ass:regular-asymptotically-linear} is satisfied by many
conventional fixed-dimensional estimators under their standard regularity
conditions. These include OLS estimators under suitable moment conditions and
uniform nonsingularity of the population regressor second-moment matrix;
linear IV and two-stage least-squares estimators under suitable moment
conditions and uniform strong identification; and repeated-cross-section or
regression difference-in-differences estimators under the corresponding
overlap, full-rank, moment, and variance-estimation conditions. More generally,
the assumption applies to regular estimators that are smooth functions of
finitely many population moments when the relevant derivatives are well
behaved, the asymptotic variances are nondegenerate, and the covariance
estimator is uniformly consistent.
\begin{corollary}\label{cor:general-estimator-uniform-validity}
Suppose Assumption \ref{ass:regular-asymptotically-linear} holds. Then, for
$\alpha\in(0,1/2)$ and $\ell\in\{m,s\}$, the LF test with the Gaussian
parametric critical value satisfies
\begin{align}\label{eq:general-estimator-LF-validity}
\limsup_{n\to\infty}
\sup_{P\in\mathbf P_{\beta,0}}
\mathbb E_P[\phi_{n,\beta}^{\ell}]
\le\alpha.
\end{align}
For any $\tau\in(0,\alpha)$, the corresponding conditional test satisfies
\begin{align}\label{eq:general-estimator-conditional-validity}
\limsup_{n\to\infty}
\sup_{P\in\mathbf P_{\beta,0}}
\mathbb E_P[\psi_{n,\beta}^\ell]
\le\alpha.
\end{align}
If, in addition,
\begin{align}\label{eq:general-estimator-eigenvalue-condition}
\inf_{P\in\mathbf P}\lambda_{\min}(\Omega(P))>0,
\end{align}
then \eqref{eq:general-estimator-LF-validity} and
\eqref{eq:general-estimator-conditional-validity} also hold for $\ell=q$.
\end{corollary}
The proof is given in Subsection~\ref{subsec:supporting-regular-estimators}. The sample-mean framework in the main text is a special case. Taking
$\beta(P)=\mathbb E_P[W_i]$, $\hat\beta_n=\bar W_n$, and
$\zeta_P(W_i)=W_i-\mu(P)$ makes
\eqref{eq:uniform-asymptotic-linearity-general} an exact identity and reduces
\eqref{eq:UI-normalized-influence-function} to
\eqref{def:uniform-integrability}. Condition
\eqref{eq:studentization-general-consistency} then follows from Lemmas
\ref{alem:consistency-sample-variance} and
\ref{alem:consistency-sample-corr}.
\section{Proof of the main results}
Throughout this section, when $\ell=q$, with a slight abuse of notation,
$\hat\Omega_n$ denotes the matrix used in the test: it denotes the sample
correlation matrix when $\hat\Omega_n\succ0$, and otherwise denotes
\[
\tilde\Omega_n
:=
\frac{\hat\Omega_n+\varepsilon_n I_k}{1+\varepsilon_n},
\qquad
\varepsilon_n>0,\quad \varepsilon_n\to0.
\]
Accordingly, every occurrence of $T_n^q$, $J^q$, and $\hat c_n^q$ is
understood to use this same matrix.
\subsection{Proof of Theorem \ref{thm:LFtest-uniform-validity}}
Here we provide the proof of the uniform asymptotic validity of the least favorable test with the parametric bootstrap critical value. The proof of the nonparametric bootstrap counter part is similar and is relegated to Section \ref{subsubsec:LF-nonparametric-bootstrap-validity}.
Suppose, by way of contradiction, that the upper bound fails for
\(\ell\in\{m,s,q\}\).
Then there exist \(\eta>0\), a subsequence \(n_l\), and a sequence
\(P_{n_l}\in\mathbf P_0\) such that
\begin{align}\label{contra-hypo:LFC-uniform-validity}
\mathbb E_{P_{n_l}}[\phi^\ell_{n_l}]
=
P_{n_l}\{T^\ell_{n_l}>\hat c^\ell_{n_l}(1-\alpha)\}
>
\alpha+\eta
\end{align}
for all \(l\). Since \(P_{n_l}\in\mathbf P_0\), after passing to a further subsequence, still denoted by
\(n_l\), we may assume that either $\mu(P_{n_l})\in\mathbf R_+^k$ for all $\ell$ or $\mu(P_{n_l})\in\mathbf R_-^k$ for all $\ell$. We prove the result for the latter. The positive case follows by replacing \(W_i\) by \(-W_i\). By compactness of \(\bar{\mathbf O}\), after passing to a further subsequence, $\Omega(P_{n_l})\to\Omega^*$ for some \(\Omega^*\in\bar{\mathbf O}\).
By extracting a further subsequence if necessary, each coordinate of $\frac{\sqrt{n_l}\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}$ has an extended limit in \([-\infty,0]\). There are two cases.
\noindent
\textbf{Case 1:} Suppose $\tfrac{\sqrt{n_l} \mu_j(P_{n_l})}{ \sigma_j(P_{n_l})} \to -\infty $ for all $j.$ By Lemma \ref{lem:statistic-degenerate-case}, $T^m_{n_l}\overset{P_{n_l}}{\to} -\infty$ and $T^\ell_{n_l}\overset{P_{n_l}}{\to}0$ for $\ell=s,q.$ We show that the least favorable critical value in \eqref{def:LF-cv-normal} is bounded away from \(-\infty\) for $\ell=m$ and from 0 for $\ell=s,q$ for any realization of $\hat{\Omega}_{n_l}.$
For $\ell=m$, the lower bound of $\hat c^m_{n_{l}}(1-\alpha)$ holds by the following:
\begin{align*}
\hat c^m_{n_{l}}(1-\alpha)\overset{(a)}{\ge} (J^m)^{-1}(1-\alpha,0_k,\hat\Omega_{n_{l}}) \overset{(b)}{\ge} \Phi^{-1}(\tfrac{1-\alpha}{2}) \qquad\text{for all }n_{l}.
\end{align*}
Here, (a) holds by the definition of $\hat c^m_{n_{l}}(1-\alpha)$. Let $\iota_{k\times k}$ is the $k\times k$ matrix whose elements are ones. For any fixed $\hat{\Omega}_{n_l}$,
\[
J^m(x,0_k,\hat\Omega_{n_{l}})
\le
J^m(x,0_k,\iota_{k\times k})= P(-|Z_1|\leq x)=2\Phi(x) I\{x<0\} + I\{x\geq 0\}
\quad\forall x\in\mathbf R,
\]
where the first inequality holds by Lemma \ref{lem:slepian-type} and the first equality holds because $T^m( \iota_{k\times k}^{1/2}Z)\overset{d}{=}{-|Z_1|}$ with $Z=(Z_1,...,Z_k)\sim N(0_k, I_k).$ Then (b) follows from the stochastic dominance of $J^m(x,0_k,\hat\Omega_{n_{l}}) $ above.
For $\ell=s,$ let $A:=(\hat\Omega_{n_l})_{-1,-1}$ be the subcorrelation matrix obtained by deleting the first row and the first column and let $Y\sim N(0_{k-1},A)$. Let $q^*>0$ be such that $\frac12
+
(k-1)\frac{\sqrt{q^*}}{\sqrt{2\pi}}
<
1-\alpha.$ Such a \(q^*\) exists because \(\alpha<1/2\). Then, for each fixed realized value of
\(\hat\Omega_{n_l}\),
\begin{align*}
\hat c^s_{n_l}(1-\alpha)
&\overset{(a)}{\ge}
\liminf_{M\to\infty}
(J^s)^{-1}(1-\alpha,(-M,0_{k-1})',\hat\Omega_{n_l})\\
&\overset{(b)}{\ge}
\inf\{
x\in\mathbf R:
P\{
\textstyle\sum_{j=2}^k Y_j^2 I\{Y_j\ge0\}\le x
\}
\ge 1-\alpha
\} \overset{(c)}{\ge}
q^*.
\end{align*}
Here, (a) holds by definition of the least favorable critical value. For (b), note $T^s(\hat{\Omega}_{n_l}^{1/2}Z+(-M,0_{k-1})')
\Rightarrow
\sum_{j=2}^k Y_{j}^2I\{Y_{j}\ge0\}$ as $M\to\infty$ for fixed $\hat{\Omega}_{n_l}$. Applying the usual lower semicontinuity of generalized
quantiles under weak convergence pathwise gives (b). For (c), note that for any \(t>0\),
\[
\begin{aligned}
P\{
\textstyle\sum_{j=2}^k Y_j^2I\{Y_j\ge0\}\le t
\}
&\le
P\{Y\in\mathbf R_-^{k-1}\}
+
\textstyle\sum_{j=2}^k P\{0<Y_j\le \sqrt t\} \le
\frac12
+
(k-1)\frac{\sqrt t}{\sqrt{2\pi}}
\end{aligned}
\]
where the second inequality holds because $\phi(0)=1/\sqrt{2\pi} \geq \phi(t)$. Then the definition of $q^*$ and $\alpha\in(0,\tfrac12)$ imply $(c).$
For $\ell=q$, the additional eigenvalue condition implies \(\Omega^*\succ0\) for $\ell=q$. On the event that $\hat\Omega_{n_l}$ is positive definite $\{\hat\Omega_{n_l}\succ0\}$, define $A:=(\hat\Omega_{n_l})_{-1,-1}$ and ${Y}\sim N(0_{k-1},A)$. Then, on such fixed $\hat\Omega_{n_l}$,
\begin{align*}
\hat c^q_{n_l}(1-\alpha)
&\ge\liminf_{M\to\infty}
(J^q)^{-1}(1-\alpha,(-M,0_{k-1})',\hat\Omega_{n_l})\\
&\overset{(a)}{\ge}
\inf\{
x\in\mathbf R:
P\{
\inf_{\nu\in\mathbf R_-^{k-1}}
(Y-\nu)'A^{-1}(Y-\nu)\le x
\}
\ge 1-\alpha
\}\\
&\overset{(b)}{\ge}
\inf\{
x\in\mathbf R:
P\{
\tfrac{1}{k-1}\textstyle\sum_{j=2}^k Y_j^2I\{Y_j\ge0\}\le x
\}
\ge 1-\alpha
\}
\overset{(c)}{\ge} q^{\dagger}>0
\end{align*}
where (a) holds because the test statistic $T^q(\hat \Omega_{n_l}^{1/2}Z+(-M,0_{k-1})', \hat\Omega_{n_l})$ reduces to $\inf_{\nu\in\mathbf R_-^{k-1}}
(Y-\nu)'A^{-1}(Y-\nu)$ as $M\to\infty$ for fixed $\hat\Omega_{n_l}$; (b) holds because
$$\inf_{\nu\in\mathbf R_-^{k-1}}
(Y-\nu)'A^{-1}(Y-\nu)\geq \inf_{\nu\in\mathbf R_-^{k-1}}
(Y-\nu)'\tfrac{1}{k-1} I_{k-1}(Y-\nu)=
\tfrac{1}{k-1}\textstyle\sum_{j=2}^k Y_j^2I\{Y_j\ge0\}$$
where the inequality follows from
$A^{-1}\succeq \tfrac{1}{\lambda_{\max}(A)}I_{k-1}\succeq\tfrac{1}{\operatorname{tr}(A)}I_{k-1}=\tfrac{1}{k-1}I_{k-1}$; and finally (c) holds similarly as above by choosing $q^{\dagger}$ small enough that $\frac12
+
(k-1)\frac{\sqrt{(k-1)q^\dagger}}{\sqrt{2\pi}}
<
1-\alpha$.
Therefore, $P_{n_l}\{T^\ell_{n_l}>\hat c^\ell_{n_l}(1-\alpha)\}\to0 $ for $\ell=m,s$. For $\ell=q$, we have
\[
\begin{aligned}
P_{n_l}\{T_{n_l}^q>\hat c_{n_l}^q\}
&\le
P_{n_l}\{\hat\Omega_{n_l}\not\succ0\}+
P_{n_l}\{T_{n_l}^q>\hat c_{n_l}^q,
\hat\Omega_{n_l}\succ0\}
\le P_{n_l}\{\hat\Omega_{n_l}\not\succ0\}+
P_{n_l}\{T_{n_l}^q>q^{\dagger}\}.
\end{aligned}
\]
Since $P_{n_l}\{\hat\Omega_{n_l}\not\succ0\}\to0$ by Lemma \ref{lem:sample-corr-positive-definite}, we have
$T_{n_l}^q\overset{P_{n_l}}{\to}0$, which contradicts \eqref{contra-hypo:LFC-uniform-validity}.
\medskip
\noindent
\textbf{Case 2:} Suppose there exists a nonempty set \(I\subset[k]\) such that
\begin{align*}
\begin{aligned}
\tfrac{\sqrt{n_l} \mu_j(P_{n_l})}{ \sigma_j(P_{n_l})} \to \delta_j \in(-\infty, 0] \quad\text{ for } j\in I,
\qquad
\tfrac{\sqrt{n_l} \mu_j(P_{n_l})}{ \sigma_j(P_{n_l})} \to -\infty
\quad\text{ for }j \not\in I.
\end{aligned}
\end{align*}
For $\ell=m,s$, Lemma \ref{lem:conv-ip-to-Jn} implies that there exists \(N=N(\eta)>0\) such that for all \(n_l\geq N\),
\begin{gather*}
P_{n_l}\{
\sup_{x\in \mathbf{R}}
|
J^\ell(x, \sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}), \hat{\Omega}_{n_l})
-
J^\ell_{n_l}(x)
|
\leq \frac{\eta}{4}
\}
\geq 1-\frac{\eta}{4}.
\end{gather*}
Then, for all \(n_l\geq N\),
\begin{gather*}
P_{n_l}\{T^\ell_{n_l}>\hat{c}^\ell_{n_l}(1-\alpha)\}
\leq
P_{n_l}\{
T^\ell_{n_l}
>
(J^\ell)^{-1}(
1-\alpha,
\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),
\hat{\Omega}_{n_l}
)
\}
\leq
\alpha+\frac{\eta}{2},
\end{gather*}
where the first inequality holds by definition of
\(\hat{c}^\ell_{n_l}\), and the second inequality holds by Lemma A.1.(vi) of
\citeauthor{RomanoShaikh2012AoS} \(\left(\citeyear{RomanoShaikh2012AoS},\right.\) henceforth RS\(\left.\right)\).
This is a contradiction to \eqref{contra-hypo:LFC-uniform-validity}.
For $\ell=q$, define $Q_{n_l}
:=
(J^q)^{-1}(
1-\alpha,
\sqrt{n_l}S_{n_l}^{-1}\mu(P_{n_l}),
\hat\Omega_{n_l}
)$ on the event $\{\hat\Omega_{n_l}\succ0\}$. By the definition of the least favorable critical value, $\hat c_{n_l}^q(1-\alpha)\ge Q_{n_l}$ on $\{\hat\Omega_{n_l}\succ0\}$. Therefore,
\[
\begin{aligned}
P_{n_l}\{
T_{n_l}^q>\hat c_{n_l}^q(1-\alpha)
\}
&\le
P_{n_l}\{\hat\Omega_{n_l}\not\succ0\}
+
P_{n_l}\{T_{n_l}^q>Q_{n_l},\,\hat\Omega_{n_l}\succ0\}
\le
P_{n_l}\{\hat\Omega_{n_l}\not\succ0\}+
\alpha+\frac{\eta}{2}
\end{aligned}
\]
where we obtain the second inequality as before. For all sufficiently large $n_l$,
$P_{n_l}\{\hat\Omega_{n_l}\not\succ0\}\le\eta/4$ by Lemma \ref{lem:sample-corr-positive-definite}, and hence
\[
P_{n_l}\{
T_{n_l}^q>\hat c_{n_l}^q(1-\alpha)
\}
\le
\alpha+\frac{3\eta}{4}
<
\alpha+\eta,
\]
contradicting \eqref{contra-hypo:LFC-uniform-validity}.
\subsection{Proof of Corollary \ref{cor:LFCtest-uniform-validity-equality}}
\begin{corollary} Suppose the assumptions of Theorem \ref{thm:LFtest-uniform-validity} and additionally that
there exists a distribution $Q$ on $\mathbf R^k$ satisfying
\[
\mu(Q)=0_k,
\qquad
\Omega(Q)=I_k,
\qquad
0<\sigma_j(Q)<\infty
\quad\text{for }j=1,\ldots,k,
\]
such that, writing $D_Q:=\operatorname{diag}
\bigl(\sigma_1(Q),\ldots,\sigma_k(Q)\bigr),$ the location family generated by $Q$ is contained in $\mathbf P$:
\begin{align}\label{ass:location-submodel}
\mathcal L(X+D_Qa)\in\mathbf P
\qquad
\text{for every }a\in\mathbf R^k,
\quad X\sim Q.
\end{align}
Then, for each $\ell\in\{m,s\}$,
\[
\limsup_{n\to\infty}
\sup_{P\in\mathbf P_0}
\mathbb E_P[\phi_n^\ell]
=
\alpha.
\]
The same conclusion holds for $\ell=q$ if, in addition, $\inf_{P\in\mathbf P}
\lambda_{\min}(\Omega(P))>0.$
\end{corollary}
\begin{proof}
By Theorem
\ref{thm:LFtest-uniform-validity}, $\limsup_{n\to\infty}
\sup_{P\in\mathbf P_0}
\mathbb E_P[\phi_n^\ell]
\leq\alpha$ for $\ell\in\{m,s\}$, and also for $\ell=q$ under the additional uniform eigenvalue condition. It remains to establish the reverse
inequality.
Let $X_1,X_2,\ldots$ be i.i.d. with distribution $Q$. Let $P_n$ be the distribution of
\[
W_{i,n}
:=
X_i+\frac{D_Q\vartheta_n}{\sqrt n},\quad \vartheta_n:=(n^{1/4},0,\ldots,0)'.
\]
By \eqref{ass:location-submodel}, $P_n\in\mathbf P$ for every $n$.
Moreover, $\mu(P_n)
=
\frac{D_Q\vartheta_n}{\sqrt n}
\in\mathbf R_+^k,$ so that $P_n\in\mathbf P_0$ for every $n.$ Since translation does not affect variances or correlations, $D(P_n)=D_Q,$ and $\Omega(P_n)=I_k,$ and $\sqrt nD(P_n)^{-1}\mu(P_n)
=
\vartheta_n.$ Thus the first coordinate of the standardized local mean diverges to
$+\infty$, while the remaining coordinates are zero.
Let $\bar X_n$ denote the sample mean of $X_1,\ldots,X_n$. Because sample variances and sample correlations are invariant to location shifts, the sample standard deviation matrix $S_n$ and sample correlation matrix $\hat\Omega_n$ computed from $W_{1,n},\ldots,W_{n,n}$ are the same as those computed from $X_1,\ldots,X_n$. Therefore, by Lemmas \ref{alem:consistency-sample-variance}-\ref{alem:uniformCLT}, $S_n^{-1}D_Q\overset{P_n}{\to}I_k,$ $\hat\Omega_n\overset{P_n} {\to}I_k$, and $D_Q^{-1}\sqrt n\bar X_n
\Rightarrow
Z$ where $Z=(Z_1,\ldots,Z_k)'\sim N_k(0_k,I_k).$ Consequently, for $\sqrt nS_n^{-1}\bar W_n
=
S_n^{-1}D_Q
(
D_Q^{-1}\sqrt n\bar X_n+\vartheta_n
)$, it follows that
\[
\frac{\sqrt n\bar W_{1,n}}{S_{1,n}}
\overset{P_n}{\longrightarrow}+\infty,\qquad
\left(
\frac{\sqrt n\bar W_{2,n}}{S_{2,n}},
\ldots,
\frac{\sqrt n\bar W_{k,n}}{S_{k,n}}
\right)'
\Rightarrow
(Z_2,\ldots,Z_k)'.
\]
Define $V_{k-1}:=\sum_{j=2}^k Z_j^2I\{Z_j\leq0\}$. Then, the convergence of test statistics are below:
\begin{align}\label{location-submodel-stat-convergence}
T_n^m\Rightarrow \max_{2\leq j\leq k}(-Z_j),
\quad
T_n^s \Rightarrow V_{k-1},
\quad
T_n^q
\Rightarrow
\inf_{\nu\in\mathbf R_+^{k-1}}
\left\|
(Z_2,\ldots,Z_k)'-\nu
\right\|^2
=V_{k-1}.
\end{align}
For $\ell=m$, the first branch in the definition of $T^m$ diverges to $+\infty$, while the second branch depends asymptotically only on coordinates $2,\ldots,k$. For $\ell=s$, the positive-part sum diverges because its first-coordinate term diverges, whereas the negative-part sum has a zero first-coordinate contribution with probability approaching one. For $\ell=q$, under the additional eigenvalue condition, the same partial-divergence argument as in Lemma \ref{lem:conv-ip-to-Jn} applies. The distance to $\mathbf R_-^k$ diverges, while the distance to $\mathbf R_+^k$ reduces to the squared distance of $(Z_2,\ldots,Z_k)'$ from $\mathbf R_+^{k-1}$. Since the limiting correlation matrix is $I_k$, Moreover, $\hat\Omega_n\succ0$ with probability approaching one, so the regularization convention for singular sample correlation matrices has no asymptotic effect.
By Lemma \ref{lem:continuity-critical-value-all}, we have the convergence below:
\begin{align}\label{eq:location-submodel-cv-convergence}
\hat c_n^\ell
=
c^\ell(1-\alpha,\hat\Omega_n)
\overset{P_n}{\to}
c^\ell(1-\alpha,I_k)
=\left\{
\begin{matrix}
\Phi^{-1}((1-\alpha)^{\frac{1}{k-1}}) & \ell=m\\
(1-\alpha)\text{-quantile of }\sum^{k-1}_{r=1}{k-1 \choose r}2^{-(k-1)}\chi^2_r & \ell=s, q
\end{matrix}\right.
\end{align}
where the critical value is from \eqref{def:LF-cv-normal}.
Note that $c^m(1-\alpha,I_k)$ is the $1-\alpha$ qunatile of $\max_{2\leq j \leq k} (-Z_j)$. Moreover, $P\{V_{k-1}=0\}
=
P\{Z_j\geq0\text{ for all }j=2,\ldots,k\}
=
2^{-(k-1)}
\leq\frac12
<
1-\alpha.$ Hence the $(1-\alpha)$-quantile of $V_{k-1}$ is strictly positive. Since the distribution of $V_{k-1}$ is continuous on $(0,\infty)$, $P\{V_{k-1}>c^\ell(1-\alpha,I_k)\}
=
\alpha$ for $\ell\in\{s,q\}.$ Combining above results with \eqref{eq:location-submodel-cv-convergence}, and using continuity of the corresponding limiting distribution at its critical value, gives $P_n\{T_n^\ell>\hat c_n^\ell\}\to \alpha.$
Since $P_n\in\mathbf P_0$ for every $n$, $\limsup_{n\to\infty}
\sup_{P\in\mathbf P_0}
\mathbb E_P[\phi_n^\ell]
\geq
\limsup_{n\to\infty}
\mathbb E_{P_n}[\phi_n^\ell]=\alpha.$ Combining this lower bound with the upper bound from Theorem \ref{thm:LFtest-uniform-validity} proves the result.
\end{proof}
\subsection{Proof of Corollary \ref{cor:LFCtest-nonnegative-cv}}
For \(P\in\mathbf P_0\), both \(\Omega(P)\) and \(I_k\) are correlation matrices. Lemma \ref{lemma:monotonicity-c}, applied with
\(\tilde\theta_2=\Omega(P)\) and \(\theta_2=I_k\), implies $c^m(1-\alpha,\Omega(P)) \leq c^m(1-\alpha,I_k).$ Consequently,
\begin{align}
\mathbb E_P[\phi_n^{m,M}] := P\{T_n^m>c^m(1-\alpha,I_k)\}
&\leq
P\left\{
T_n^m>c^m(1-\alpha,\Omega(P))
\right\}.
\label{eq:fixed-cv-oracle-bound}
\end{align}
Under the assumptions of Theorem
\ref{thm:LFtest-uniform-validity}, the argument in its proof also gives uniform validity of the infeasible oracle test that uses the population correlation matrix:
\[
\limsup_{n\to\infty}
\sup_{P\in\mathbf P_0}
P\left\{
T_n^m>c^m(1-\alpha,\Omega(P))
\right\}
\leq \alpha.
\]
Taking the supremum over \(P\in\mathbf P_0\) in
\eqref{eq:fixed-cv-oracle-bound} and then the limit superior yields
\[
\limsup_{n\to\infty}
\sup_{P\in\mathbf P_0}
\mathbb E_P[\phi_n^{m,M}]
\leq \alpha.
\]
\subsection{Proof of Theorem \ref{thm:conditional-uniform-validity}}
Suppose, by way of contradiction, that the conclusion fails. Then there exist
$\eta>0$, a subsequence $n_l$, and a sequence
$\{P_{n_l}\}\subset\mathbf P_0$ such that
\begin{align}\label{eq:conditional-validity-contraposition}
\mathbb E_{P_{n_l}}[\psi_{n_l}] > \alpha+\eta
\qquad\text{for all }n_l.
\end{align}
By Bolzano-Weierstrass theorem, there exists a further subsequence, still denoted by $n_l$, such that $\Omega(P_{n_l})\to \Omega^* \in \bar{\mathbf{O}}$. By Lemma \ref{alem:consistency-sample-corr} and the triangle inequality, $\|\hat{\Omega}_{n_l} -\Omega^*\|\overset{P_{n_l}}{\to}0$. By continuity of $\kappa_\tau(\Omega)$ in $\bar{\mathbf{O}}$ by Lemma \ref{lem:kappa-continuity} and the continuous mapping theorem,
\begin{align*}
\hat{\kappa}_{n_l} := \kappa_\tau (\hat{\Omega}_{n_l}) \overset{P_{n_l}}{\to}\kappa_\tau ({\Omega}^*) =:\kappa^*
\end{align*}
By symmetry of the null and of the test construction, it is enough to consider the case $\mu(P_{n_l})\in \mathbf R_+^k$ for all ${n_l}$. As in the proof for Theorem \ref{thm:LFtest-uniform-validity}, we now distinguish two cases.
\noindent
\textbf{Case 1:}
\[
\frac{\sqrt {n_l}\,\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}\to +\infty
\qquad\forall j\in[k].
\]
By Lemma \ref{lem:statistic-degenerate-case}, we have $ \sqrt{n_l} \bar W_{j,n_l} /S_{j,n_l}\overset{P_{n_l}}{\to} \infty$ for all $j\in[k].$ The union bound gives
\[
P_{n_l} \{\mathcal I_+=[k]\} = P_{n_l} \{ \min_{1\le j\le k} \sqrt{{n_l}} \bar W_{j,{n_l}} /S_{j,{n_l}}> \hat{\kappa}_{n_l} \}
\ge
1-\textstyle\sum_{j=1}^k P_{n_l} \{\sqrt{{n_l}} \bar W_{j,{n_l}} /S_{j,{n_l}}\le \hat{\kappa}_{n_l}\}
\to 1.
\]
Since $ I_{\mathrm{alt}} + I_{\mathrm{pos}} +I_{\mathrm{neg}}+I_{\mathrm{origin}}= 1-I_{null}$ and $P_{n_l}\{I_{null} =1\} = P_{n_l}\{\mathcal{I}_+=[k] \text{ or }\mathcal{I}_-=[k]\} {\to}1$, we have
\[
\mathbb E_{P_{n_l}}[\psi_{n_l}]\le 1- P_{n_l} \{I_{null}=1\} \to 0,
\]
contradicting \eqref{eq:conditional-validity-contraposition}.
\noindent
\textbf{Case 2:}
There exists a further subsequence and a nonempty set $I\subset[k]$ such that
\[
\frac{\sqrt {n_l}\,\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}\to \delta_j\in[0,\infty)\quad
\text{for}\quad j\in I\quad\text{and}\quad
\frac{\sqrt {n_l}\,\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}\to +\infty
\quad
\text{for}\quad j\not\in I.
\]
Any realization with $I_{\mathrm{alt}}=1$ or
$I_{\mathrm{neg}}=1$ must contain at least one coordinate below $-\hat\kappa_{n_l}$. Hence,
with
\[
B_{-,n_l}:=\{\exists j\in[k]: \sqrt{n_l} \bar W_{j,n_l} /S_{j,n_l}<-\hat\kappa_{n_l}\},
\]
we have $\{I_{\mathrm{alt}}=1\}\cup\{I_{\mathrm{neg}}=1\}\subseteq B_{-,n_l}.$ Therefore,
\begin{align}
\mathbb E_{P_{n_l}}[\psi_{n_l}]
&=
P_{n_l}\{I_{\mathrm{alt}}=1\}
+
P_{n_l} \{I_{\mathrm{neg}}=1,\psi_{n_l}^-(\alpha-\tau)=1\}
\nonumber\\
&+
\mathbb E_{P_{n_l}}[ I_{\mathrm{pos}}\,\psi_{n_l}^+(\alpha-\tau)]
+
\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\,\psi_{n_l}^0(\alpha-\tau)] \nonumber \\
&\le
P_{n_l}\{B_{-,{n_l}}\}
+
\mathbb E_{P_{n_l}}[ I_{\mathrm{pos}}\,\psi_{n_l}^+(\alpha-\tau)]
+
\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\,\psi_{n_l}^0(\alpha-\tau)].\label{eq:main-decomp}
\end{align}
First we show
\begin{align}\label{eq:B-op(1)}
\limsup_{{n_l}\to\infty} P_{n_l}\{B_{-,{n_l}}\}\le \tau .
\end{align}
Without loss of generality, write $I=\{1,\dots,\tilde k\}$. By Lemmas \ref{alem:consistency-sample-variance} and \ref{alem:uniformCLT}, $(\sqrt{n_l}\bar W_{j,n_l}/S_{j,n_l} )_{j\in I} \Rightarrow N(\delta_I,\Omega^*_{I\times I})$
where $\delta_I=(\delta_j)_{j\in I}$ and $\Omega^*_{I\times I}$ is the principal
submatrix of $\Omega^*$ indexed by $I$.
Also, for every $j\notin I$, $ \sqrt{n_l} \bar W_{j,n_l} /S_{j,n_l}\overset{P_{n_l}}{\to} \infty$. Hence
\[
P_{n_l}\{B_{-,n_l}\}
\le
P_{n_l} \{\min_{j\in I}\sqrt{n_l} \bar W_{j,n_l} /S_{j,n_l}<-\hat\kappa_{n_l}\}
+
P_{n_l}\{\exists j\notin I: \sqrt{n_l} \bar W_{j,n_l} /S_{j,n_l} <-\hat\kappa_{n_l}\},
\]
and the second term converges to $0$. For the first term, we have
\[
\lim_{n_l\to\infty} P_{n_l} \{\min_{j\in I} \sqrt{n_l} \bar W_{j,n_l} /S_{j,n_l} <- \hat\kappa_{n_l}\}
\overset{(a)}{=}
P \{\min_{j\in I}(Y_j+\delta_j) < - \kappa^*\}
\overset{(b)}{\leq}
P \{\min_{j\in I}Y_j < - \kappa^*\}
\]
with $Y\sim N(0_I,\Omega^*_{I\times I})$ where the continuous mapping theorem and Slutsky's theorem give (a), and (b) holds because $\delta_j\ge 0$ for all $j\in I$. Furthermore,
\begin{align*}
P \{\min_{j\in I}Y_j < -\kappa^*\}
&\overset{(a)}{\leq}
P \{\max_{j\in I} Y_j > \kappa^*\}
\overset{(b)}{\leq}
P \{\max_{j\in I} (\Omega^{*1/2}Z)_j> \kappa^*\} \\
&\overset{(c)}{\leq}
P \{\max_{j\in [k]} (\Omega^{*1/2}Z)_j> \kappa^*\}
\overset{(d)}{\leq}
\tau
\end{align*}
where (a) by symmetry of mean-zero normal random variable, (b) follows from $\max_{j\in I} Y_j\overset{d}{=}\max_{j\in I} (\Omega^{*1/2}Z)_j$, (c) holds as $\max_{j\in [k]}(\Omega^{*1/2}Z)_j\geq \max_{j\in I}(\Omega^{*1/2}Z)_j$ a.s., and (d) follows from the definition of $\kappa^*.$ Therefore, we have \eqref{eq:B-op(1)}.
Next we show
\begin{align}\label{eq:conditional-test-validity-nondegenerate}
\limsup_{l\to\infty} \{ \mathbb E_{P_{n_l}}[ I_{\mathrm{pos}}\,\psi_{n_l}^+(\alpha-\tau)]
+
\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\,\psi_{n_l}^0(\alpha-\tau)]\} \leq \alpha-\tau
\end{align}
To this end, define
\begin{align*}
a_{n_l} &:=
\begin{cases}
\mathbb E_{P_{n_l}}\left[\psi_{n_l}^+(\alpha-\tau)\mid I_{\mathrm{pos}}=1\right],
& \text{if } P_{n_l}\{I_{\mathrm{pos}}=1\}>0,\\[4pt]
0, & \text{if } P_{n_l}\{I_{\mathrm{pos}}=1\}=0,
\end{cases} \text{ and }\\
b_{n_l} &:=
\begin{cases}
\mathbb E_{P_{n_l}}\left[\psi_{n_l}^0(\alpha-\tau)\mid I_{\mathrm{origin}}=1\right],
& \text{if } P_{n_l}\{I_{\mathrm{origin}}=1\}>0,\\[4pt]
0, & \text{if } P_{n_l}\{I_{\mathrm{origin}}=1\}=0.
\end{cases}
\end{align*}
Then, for every \(n_l\), $\mathbb E_{P_{n_l}}[I_{\mathrm{pos}}\psi_{n_l}^+(\alpha-\tau)]
=
a_{n_l}\,P_{n_l}\{I_{\mathrm{pos}}=1\},$ and $\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\psi_{n_l}^0(\alpha-\tau)]
=
b_{n_l}\,P_{n_l}\{I_{\mathrm{origin}}=1\}$. Since \([0,1]^2\) is compact, after passing to a further subsequence if necessary, we may
assume
\[
P_{n_l}\{I_{\mathrm{pos}}=1\}\to p_+ \in[0,1],
\qquad
P_{n_l}\{I_{\mathrm{origin}}=1\}\to p_0\in[0,1].
\]
We now bound the two terms separately.
If \(p_+=0\), then $\mathbb E_{P_{n_l}}[I_{\mathrm{pos}}\psi_{n_l}^+(\alpha-\tau)]
\le P_{n_l}\{I_{\mathrm{pos}}=1\}\to 0.$ If \(p_+>0\), then \(\liminf_{n_l\to\infty}P_{n_l}\{I_{\mathrm{pos}}=1\}>0\), and
Lemma \ref{lem:branchwise-conditional-approx} implies that for any \(\varepsilon>0\),
there exists \(L_1=L_1(\varepsilon)\) such that for all \(n_l\ge L_1\), $a_{n_l}\le \alpha-\tau+\varepsilon.$ Hence, for all \(n_l\ge L_1\),
\[
\mathbb E_{P_{n_l}}[I_{\mathrm{pos}}\psi_{n_l}^+(\alpha-\tau)]
\le
(\alpha-\tau+\varepsilon)P_{n_l}\{I_{\mathrm{pos}}=1\}.
\]
Similarly, if \(p_0=0\), then $\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\psi_{n_l}^0(\alpha-\tau)]
\le P_{n_l}\{I_{\mathrm{origin}}=1\}\to 0.$ If \(p_0>0\), then necessarily \(I=[k]\).
Lemma \ref{lem:branchwise-conditional-approx} implies that for any
\(\varepsilon>0\), there exists \(L_2=L_2(\varepsilon)\) such that for all \(n_l\ge L_2\), $b_{n_l}\le \alpha-\tau+\varepsilon.$ Hence, for all \(n_l\ge L_2\),
\[
\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\psi_{n_l}^0(\alpha-\tau)]
\le
(\alpha-\tau+\varepsilon)P_{n_l}\{I_{\mathrm{origin}}=1\}.
\]
Combining the two cases, we obtain that for any \(\varepsilon>0\),
\[
\limsup_{n_l\to\infty}
\Big(
\mathbb E_{P_{n_l}}[I_{\mathrm{pos}}\psi_{n_l}^+(\alpha-\tau)]
+
\mathbb E_{P_{n_l}}[I_{\mathrm{origin}}\psi_{n_l}^0(\alpha-\tau)]
\Big)
\le
(\alpha-\tau+\varepsilon)(p_+ + p_0).
\]
Since \(I_{\mathrm{pos}}\) and \(I_{\mathrm{origin}}\) are disjoint events, \(p_+ + p_0\le 1\). Therefore we have \eqref{eq:conditional-test-validity-nondegenerate}. Combining this bound with \eqref{eq:main-decomp} and \eqref{eq:B-op(1)}, we obtain
\[
\limsup_{{n_l}\to\infty}\mathbb E_{P_{n_l}}[\psi_{n_l}]
\le
\alpha+\varepsilon.
\]
Since \(\varepsilon>0\) is arbitrary, it follows that $\limsup_{{n_l}\to\infty}\mathbb E_{P_{n_l}}[\psi_{n_l}]
\le \alpha,$ which contradicts \eqref{eq:conditional-validity-contraposition}. The case $\mu(P_n)\in\mathbf R_-^k$ is identical after exchanging the roles of the positive and negative branches. The proof is complete.
\subsection{Proof of conditional critical values}
\begin{proposition}
\label{prop:conditional-lf-pos-branch}
Fix a realization of the first-step screening sets \(\mathcal I_+\), \(\mathcal I_-\), and \(\mathcal I_0\), and let \(\kappa_\tau:=\kappa_\tau(I_k)\) be the threshold defined in \eqref{def:kappa-screening-threshold}. Then the following statements hold.
\begin{enumerate}
\item[(i)] For $\ell=m$ and every
$t\in[-\kappa_\tau,\kappa_\tau]$, and for $\ell\in\{s,q\}$ and every $t\geq0$, the conditional CDF on the positive branch, $L^{\ell,+}(t;\mu,I_k, \mathcal{I}_0),$
is weakly increasing in each \(\mu_j\), \(j\in\mathcal I_0\). Consequently,
\begin{align*}
\inf_{\mu\in\mathbf R_+^k}
L^{\ell,+}_{\mathcal I_0}(t,\mu,I_k)
&=
L^{\ell,+}_{\mathcal{I}_0}(t,0_k,I_k)
=\left\{
\begin{matrix}
\left(\frac{\Phi(\kappa_\tau)-\Phi(-t)}{2\Phi(\kappa_\tau)-1}\right)^{|\mathcal{I}_0|}& \text{ for }\ell=m\\
\sum_{r=0}^{|\mathcal{I}_0|} {|\mathcal{I}_0|\choose r}2^{-|\mathcal{I}_0|} F_{\kappa_\tau}^{*r}(t) & \text{ for }\ell=s,q
\end{matrix}\right.
\end{align*}
where \(F_{\kappa_\tau}^{*0}(t):=I\{t\ge 0\}\), \(F_{\kappa_\tau}^{*r}\) denotes the \(r\)-fold convolution CDF of \(F_{\kappa_\tau}\), and \(F_{\kappa_\tau}\) is the CDF on \([0,\kappa_\tau^2]\) given by
\begin{align}
F_{\kappa_\tau}(u)
:=
P\{
\chi_1^2\le u \mid \chi_1^2\le \kappa_\tau^2\}
=
\frac{2\Phi(\sqrt u)-1}{2\Phi(\kappa_\tau)-1},
\qquad 0\le u\le \kappa_\tau^2.
\label{eq:Fkappa-def}
\end{align}
\item[(ii)] For $\ell=m$ and every
$t\in[-\kappa_\tau,\kappa_\tau]$, and for $\ell\in\{s,q\}$ and every $t\geq0$, the conditional CDF on the negative branch,
\(L^{\ell,-}_{\mathcal I_0}(t,\mu,I_k)\), is weakly decreasing in each
\(\mu_j\), \(j\in\mathcal I_0\). Consequently, for all $\ell\in\{m,s,q\},$
\begin{align*}
\inf_{\mu\in\mathbf R_-^k}
L^{\ell,-}_{\mathcal I_0}(t,\mu,I_k)
=L^{\ell,-}_{\mathcal I_0}(t,0_k,I_k)
=L^{\ell,+}_{\mathcal I_0}(t,0_k,I_k).
\end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
We prove (i) first. The proof of (ii) follows by symmetry. Recall that on the positive branch, $\mathcal I_+\neq\varnothing,$ and $\mathcal I_-=\varnothing$. We first record a useful monotonicity fact. For $Z_1~N(0,1)$ and $\mu_1\in\mathbf{R}$, it is known that the truncated Gaussian random variable $Z_1+\mu_1 \mid |Z_1+\mu_1|\le \kappa_\tau$ is stochastically increasing in \(\mu_1\) by Lemma A.1 in \cite{lee2016AoS-post-selection-LASSO}. Below, we consider $\ell=m$ and $\ell=s,q$ separately.
\medskip
\noindent\textit{Case $\ell=m$.} For fixed \(t\in[-\kappa_\tau,\kappa_\tau]\), the conditional CDF can be expressed as
\begin{align*}
L^{m,+}_{\mathcal I_0}(t,\mu,I_k)
&= P\{ \max_{j\in\mathcal{I}_0} (-Z_j-\mu_j)\leq t \mid Z_j+\mu_j>\kappa_\tau ~\forall j\in\mathcal{I}_+, ~ |Z_j+\mu_j|\leq\kappa_\tau ~\forall j\in\mathcal{I}_0\}\\
&=\textstyle\prod_{j\in \mathcal I_0} P\{Z_j+\mu_j\geq -t \mid |Z_j+\mu_j|\leq\kappa_\tau\} = \textstyle\prod_{j\in \mathcal I_0} \ell_t(\mu_j)
\end{align*}
where $Z_1,...,Z_k \sim i.i.d.~N(0,1)$ and
$$\ell_t(\mu_1)
:=
1-P\{
Z_1+\mu_1 \leq -t
\mid
-\kappa_\tau\le Z_1+\mu_1\le \kappa_\tau
\}.$$
Since $Z_1+\mu_1 \mid |Z_1+\mu_1|\le \kappa_\tau
$ is stochastically increasing in \(\mu_1\), \(\ell_t(\mu_1)\) is weakly increasing in \(\mu_1\). As the positive-branch null imposes \(\mu_j\ge 0\) for all \(j\in\mathcal I_0\), the infimum is attained at the boundary point \(\mu_j=0\) for all \(j\in\mathcal I_0\). Furthermore,
\begin{align*}
L^{m,+}_{\mathcal I_0}(t,0_k,I_k)
&= \ell_t(0)^{|\mathcal{I}_0|}
= \left(\frac{\Phi(\kappa_\tau)-\Phi(-t)}{\Phi(\kappa_\tau)-\Phi(-\kappa_\tau)}\right)^{|\mathcal{I}_0|}
= \left(\frac{\Phi(\kappa_\tau)-\Phi(-t)}{2\Phi(\kappa_\tau)-1}\right)^{|\mathcal{I}_0|}.
\end{align*}
\medskip
\noindent\textit{Case $\ell=s,q$.} For fixed \(t\in[-\kappa_\tau,\kappa_\tau]\), the conditional CDFs satisfy
\begin{align*}
L^{s,+}_{\mathcal I_0}(t,\mu,I_k)
&=L^{q,+}_{\mathcal I_0}(t,\mu,I_k)\\
&= P\{ \textstyle\sum_{j\in\mathcal{I}_0} (Z_j+\mu_j)^2 I\{Z_j+\mu_j\leq 0\}\leq t
\mid |Z_j+\mu_j|\leq\kappa_\tau ~\forall j\in\mathcal{I}_0\}.
\end{align*}
Since the mapping $x\mapsto x^2I\{x\leq 0\}$ is nonincreasing, $(Z_j+\mu_j)^2 I\{Z_j+\mu_j\leq 0\}
\mid |Z_j+\mu_j|\leq\kappa_\tau$ is stochastically nonincreasing in $\mu_j$. By independence across \(j\in\mathcal I_0\), their sum $\sum_{j\in\mathcal{I}_0}(Z_j+\mu_j)^2 I\{Z_j+\mu_j\leq 0\}$ conditional on $\{|Z_j+\mu_j|\leq\kappa_\tau ~~\forall j\in\mathcal{I}_0\}$ is also nonincreasing. Therefore, $L^{\ell,+}_{\mathcal I_0}(t,\mu,I_k)$ is weakly increasing in each $\mu_j$, $j\in\mathcal{I}_0$ for both $\ell=s, q$ and the infimum is attained at the boundary point \(\mu_j=0\) for all \(j\in\mathcal I_0\).
It remains to compute the CDF at \(\mu=0_k\). Let $Y_j:=Z^2_j I\{Z_j\leq 0\} \mid |Z_j|\le \kappa_\tau$ for $j=1,...,k.$ Note that \(Y_j\) admits the mixture representation $Y_j\overset d= B U$ where \(B\sim \mathrm{Bernoulli}(\tfrac12)\), \(U\sim \chi_1^2\mid \chi_1^2\le \kappa_\tau^2\), and \(B\) and \(U\) are independent. The CDF of \(U\) is
\[
F_{\kappa_\tau}(u)
=
P\{\chi_1^2\le u\mid \chi_1^2\le\kappa_\tau^2\}
=
\frac{2\Phi(\sqrt u)-1}{2\Phi(\kappa_\tau)-1},
\qquad 0\le u\le\kappa_\tau^2.
\]
Therefore, if exactly \(r\) of the \(|\mathcal I_0|\) Bernoulli variables equal one, which
occurs with probability
\(
{|\mathcal I_0|\choose r}2^{-|\mathcal I_0|},
\)
then the sum \(\sum_{j\in\mathcal I_0}Y_j\) has CDF \(F_{\kappa_\tau}^{*r}\), with the
convention \(F_{\kappa_\tau}^{*0}(t)=I\{t\ge0\}\). Summing over
\(r=0,\ldots,|\mathcal I_0|\) gives
\[
L^{s,+}_{\mathcal I_0}(t, 0_k,I_k)
=
L^{q,+}_{\mathcal I_0}(t,0_k,I_k)
=
\textstyle\sum_{r=0}^{|\mathcal I_0|}
{|\mathcal I_0|\choose r}2^{-|\mathcal I_0|}
F_{\kappa_\tau}^{*r}(t).
\]
We now prove (ii). Let \(\tilde Z:=-Z\) and \(\tilde\mu:=-\mu\). Since
\(\tilde Z\sim N(0,I_k)\), the negative-branch conditioning event
$\{Z_j+\mu_j<-\kappa_\tau ~ \forall j\in\mathcal I_-,~
|Z_j+\mu_j|\le\kappa_\tau ~ \forall j\in\mathcal I_0\}$ is transformed into the positive-branch conditioning event
$\{\tilde Z_j+\tilde\mu_j>\kappa_\tau ~ \forall j\in\mathcal I_-,
~|\tilde Z_j+\tilde\mu_j|\le\kappa_\tau ~ \forall j\in\mathcal I_0\}$.
Moreover, for \(\Omega=I_k\), the negative-branch statistics transform into the corresponding
positive-branch statistics:
\(
T^{\ell,-}(Z+\mu, I_k,\mathcal I_0)
=
T^{\ell,+}(\tilde Z+\tilde\mu, I_k, \mathcal I_0),
\) for all $\ell.$ Hence
\[
L^{\ell,-}_{\mathcal I_0}(t,\mu,I_k)
=
L^{\ell,+}_{\mathcal I_0}(t,-\mu,I_k),
\qquad \ell\in\{m,s,q\}
\]
and the rest follows.
\end{proof}
\begin{proposition}\label{prop:conditional-lf-origin-m}
Suppose \(k\ge 2\) and \(\Omega=I_k\). Let $\kappa_\tau:=\kappa_\tau(I_k)$ and, for each \(r\ge 0\), define $\theta_1^{(r)}:=(r,0,\ldots,0)' \in \mathbf R^k .$ Consider the origin-branch conditional CDF $L^{m,0}(t,\mu,I_k)
$ in \eqref{def:Lzero-selective}. The following statements hold.
\begin{enumerate}
\item[(i)] For each fixed \(t\in[0,\kappa_\tau)\), the map \(r\mapsto L^{m,0}(t,\theta_1^{(r)},I_k)\) is weakly decreasing. Consequently, for every \(p\in(2^{1-k},1)\), the quantile $(L^{m,0})^{-1}(p,\theta_1^{(r)},I_k)
=
\inf\{t\in\mathbf R:
L^{m,0}(t,\theta_1^{(r)},I_k)\ge p\}$ is weakly increasing in \(r\).
\item[(ii)] For $\gamma\in(0,1-2^{1-k}),$
\begin{align}\label{eq:max-stat-lf}
\begin{aligned}
\sup_{\mu\in\mathbf{R}^k_+\cup\mathbf{R}^k_-}(L^{m,0})^{-1}(1-\gamma,\mu,I_k) &=\lim_{r\to\infty}(L^{m,0})^{-1}(1-\gamma,\theta_1^{(r)},I_k) \\
&= \Phi^{-1}\left(1-\Phi(\kappa_\tau)+\bigl(2\Phi(\kappa_\tau)-1\bigr)(1-\gamma)^{1/(k-1)}
\right).
\end{aligned}
\end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
Fix \(t\in[0,\kappa_\tau)\). For \(u\in\mathbf R\) and $Z_u\sim N(u,1)$, define $A_t(u)
=
P\{Z_u \le t \mid |Z_u|\le \kappa_\tau\}$, $B_t(u)
=
P\{Z_u\ge -t \mid |Z_u|\le \kappa_\tau\},$ and $C_t(u)
=
P\{|Z_u|\le t \mid |Z_u|\le \kappa_\tau\}$ as in the proof of Lemma \ref{lem:origin-schur-k} with $\kappa$ is replaced with $\kappa_\tau$. By applying inclusion-exclusion, write $L^{m,0}(t,\theta_1^{(r)},I_k)$ as
\begin{align}\label{eq:origin-branch-factorization}
\begin{aligned}
L^{m,0}(t,\theta_1^{(r)},I_k)
&=
\textstyle\prod_{j=1}^k A_t(\theta_{1,j}^{(r)})
+
\textstyle\prod_{j=1}^k B_t(\theta_{1,j}^{(r)})
-
\textstyle\prod_{j=1}^k C_t(\theta_{1,j}^{(r)})\\
&\overset{(a)}{=}(A_t(r)+ B_t(r) )A_t(0)^{k-1}
-
C_t(r) C_t(0)^{k-1}\\
&\overset{(b)}{=} A_t(0)^{k-1} + (A_t(0)^{k-1}- C_t(0)^{k-1} )C_t(r)
\end{aligned}
\end{align}
where (a) holds because $A_t(0)= B_t(0)$ and (b) holds because $A_t(r) + B_t(r) -1 = C_t(r).$ Because $k\geq 2$ and $A_t(0)-C_t(0)=P\{Z_0<-t\mid |Z_0|\leq \kappa_\tau\}>0$, it follows that $A_t(0)^{k-1}- C_t(0)^{k-1}>0$. Therefore, it suffices to show the monotonicity of \(C_t(r)\) in $r$.
If \(t=0\), then \(C_0(r)=0\) for all \(r\ge0\), and hence $L^{m,0}(0,\theta_1^{(r)},I_k)=A_0(0)^{k-1}=2^{1-k}$ and the desired monotonicity is trivial. Fix \(0<t<\kappa_\tau\). For $0\leq r_1 < r_2$, choose $\lambda:= \tfrac{1+r_1/r_2}{2}\in(0,1),$ and $r_1=\lambda r_2 + (1-\lambda)(-r_2).$ Since the map $r\mapsto C_t(r)$ is log-concave by Lemma~\ref{lem:conditional-interval-logconcavity} and \(C_t(r)=C_t(-r)\), we obtain
$$\log C_t(r_1) \geq \lambda \log C_t(r_2) + (1-\lambda) \log C_t(-r_2)=\log C_t(r_2).$$
Therefore, \(C_t(r)\) is weakly decreasing in \(r\geq 0\).
For (ii), recall Lemma \ref{lem:origin-schur-k} implies, for any $\mu\in\mathbf{R}^k_+$ and $t\geq0$,
$$L^{m,0} (t,\mu,I_k)\geq L^{m,0} (t,(\textstyle\sum_j \mu_j,0,...,0),I_k) \geq \lim_{r\to\infty}L^{m,0} (t,(r ,0,...,0),I_k)$$
where the second inequality is from (i). The symmetry $L^{m,0} (t,\mu,I_k)=L^{m,0} (t,-\mu,I_k)$ then gives
$$\inf_{\mu\in\mathbf{R}^k_+\cup\mathbf{R}^k_-}L^{m,0} (t,\mu,I_k) = \lim_{r\to\infty}L^{m,0} (t,(r ,0,...,0),I_k).$$ Since $L^{m,0}(0, \theta^{(r)}_1, I_k)=2^{1-k}$ and $L^{m,0}(\kappa_\tau, \theta^{(r)}_1, I_k)=1$, the first equality in \eqref{eq:max-stat-lf} follows from the lower envelope distribution. To derive the second equality in \eqref{eq:max-stat-lf}, note for any $\varepsilon>0$,
\[
C_{\kappa_\tau-\varepsilon}(r):=P\{
|Z_r|\le \kappa_\tau-\varepsilon
\mid
|Z_r|\le \kappa_\tau
\}
\to 0\quad \text{as }r\to\infty.
\]
For any $t\in[0,\kappa_\tau]$, $\lim_{r\to\infty}C_t(r)= I\{t=\kappa_\tau\}.$ The expansion in \eqref{eq:origin-branch-factorization} implies
\begin{align*}
L^{m,0}\left(t,(\infty,0,\ldots,0)',I_k\right)
=A_t(0)^{k-1}
=\left(
\frac{\Phi(t)-\Phi(-\kappa_\tau)}
{\Phi(\kappa_\tau)-\Phi(-\kappa_\tau)}
\right)^{k-1}
\end{align*}
because $A_{\kappa_\tau}(0)=C_{\kappa_\tau}(0).$ The quantile can be obtained by solving this with respect to $1-\gamma.$
\end{proof}
\begin{proposition}
\label{prop:conditional-lf-origin-sq}
Suppose \(\Omega=I_k\), and write \(\kappa:=\kappa_\tau(I_k)\). For
\(a=0,\ldots,k\), define
\[
F^{(a)}_{k,\kappa}(t)
:=
\sum_{r=0}^{k-a}
{k-a\choose r}2^{-(k-a)}
P\left\{
\min\left[
a\kappa^2+\sum_{i=1}^{k-a-r}U_i',
\sum_{i=1}^{r}U_i
\right]\le t
\right\},
\]
where \(U_i,U_i'\) are i.i.d. with distribution
\[
F_\kappa(u)
=
P\{\chi_1^2\le u\mid \chi_1^2\le\kappa^2\}
=
\frac{2\Phi(\sqrt u)-1}{2\Phi(\kappa)-1},
\qquad 0\le u\le \kappa^2.
\]
Then, for every \(t\ge0\),
\[
\inf_{\mu\in\mathbf R_+^k\cup\mathbf R_-^k}
L^{s,0}(t,\mu,I_k)
=
\min_{a=0,\ldots,k}F^{(a)}_{k,\kappa}(t).
\]
Since \(T^q(\cdot,I_k)=T^s(\cdot)\), the same conclusion holds for
\(\ell=q\).
\end{proposition}
\begin{proof}
Let \(Z \sim N(0_k, I_k)\) and $\mu\in\mathbf{R}^k_+$. Define the survival probability
\[
S_t(\mu):=
P\{T^s(Z+\mu)>t\mid |Z_j+\mu_j|\le\kappa,\ j\in[k]\}.
\]
It suffices to prove
\[
\sup_{\mu\in\mathbf R_+^k}S_t(\mu)
=
\max_{a=0,\ldots,k}
\{1-F^{(a)}_{k,\kappa}(t)\}.
\]
To this end, for any $\mu_1\geq0$, define a random variable $X_j$ and its density
$$X_j:=X_j(\mu_j) \overset{d}{=} Z_j+\mu_j \mid |Z_j+\mu_j| \leq \kappa \quad\forall j,\qquad f_{X(\mu_1)} (x):= \frac{e^{\mu_1x}\phi(x)I\{|x|\le\kappa\}}
{\int_{-\kappa}^{\kappa}e^{\mu_1x}\phi(x)\,dx}.$$
For \(x\in[-\kappa,\kappa]\), define
\begin{align*}
h(x)
&:=
P\{T^s(x,X_2,\ldots,X_k)>t\mid X_1=x\} = P\{T^s(x,X_2,\ldots,X_k)>t\}
\end{align*}
where the second equality holds because $X_j$'s are independent. Note that $h$ is V-shaped. Indeed, the function can be written as
\begin{align*}
h(x)&=\left\{
\begin{matrix}
P\{ \textstyle\sum^k_{j=2} X^2_jI \{X_j\geq0\}>t-x^2, \ \textstyle\sum^k_{j=2} X^2_jI \{X_j\leq0\}>t\} & \text{ for }x\geq0\\
P\{ \textstyle\sum^k_{j=2} X^2_jI \{X_j\geq0\}>t, \ \textstyle\sum^k_{j=2} X^2_jI \{X_j\leq0\}>t-x^2\} & \text{ for }x\leq0
\end{matrix}\right..
\end{align*}
Hence \(h(x)\) is weakly increasing on \([0,\kappa]\) and weakly decreasing on
\([-\kappa,0]\).
We claim that, for every \(\mu_1\ge0\),
\begin{align}\label{eq:end-point}
\mathbb E[h(X_1(\mu_1))]
\le
\max\{ \mathbb E[h(X_1(0))],h(\kappa)\}.
\end{align}
Put
\[
c:=\max\{\mathbb E[h(X_1(0))],h(\kappa)\},
\qquad
g(x):=h(x)-c.
\]
Since \(h\) is increasing on \([0,\kappa]\) and \(h(\kappa)\le c\), we have
\(g(x)\le0\) for all \(x\in[0,\kappa]\). Since \(h\) is decreasing on
\([-\kappa,0]\), the function \(g\) has at most one sign change on
\([-\kappa,\kappa]\), and if the sign change occurs it is from \(+\) to \(-\). Let \(x_0\) be a sign-change point of \(g\). Then,
\[
\begin{aligned}
\int_{-\kappa}^{\kappa}g(x)e^{\mu_1 x}\phi(x)\,dx
\overset{(a)}{\le}
e^{\mu_1x_0}
\int_{-\kappa}^{x_0}g(x)\phi(x)\,dx
+
e^{\mu_1x_0}
\int_{x_0}^{\kappa}g(x)\phi(x)\,dx
&=
e^{\mu_1x_0}\int_{-\kappa}^{\kappa}g(x)\phi(x)\,dx
\end{aligned}
\]
where (a) holds because \(g(x)\ge0\) for
\(x\le x_0\) and \(g(x)\le0\) for \(x\ge x_0\) and \(e^{\mu_1 x}\) is increasing in
\(x\) for \(\mu_1\ge0\). Furthermore, note that
$$ \int_{-\kappa}^{\kappa}g(x)\phi(x)\,dx=\int_{-\kappa}^{\kappa}h(x)\phi(x)\,dx-c\int_{-\kappa}^{\kappa}\phi(x)\,dx = \{\mathbb E[h(X_1(0))] -c\} \int_{-\kappa}^{\kappa}\phi(x)\,dx\leq0.$$
Dividing by the positive normalizing constant gives the claim:
\begin{align*}
\mathbb E[h(X_1(\mu_1))] -c= \int_{-\kappa}^{\kappa}g(x)f_{X(\mu_1)}(x)\,dx = \frac{\int_{-\kappa}^{\kappa}g(x)e^{\mu_1 x}\phi(x)\,dx
}{\int_{-\kappa}^{\kappa}e^{\mu_1 x}\phi(x)\,dx}\leq 0.
\end{align*}
Applying \eqref{eq:end-point} to the first coordinate gives
\[
S_t(\mu_1,\mu_2,\ldots,\mu_k)
\le
\max\left\{
S_t(0,\mu_2,\ldots,\mu_k),
S_t(\infty,\mu_2,\ldots,\mu_k)
\right\},
\]
where $S_t(\infty,\mu_2,\ldots,\mu_k):=\lim_{\mu_1\to\infty} \mathbb E[h(X_1(\mu_1))]=h(\kappa)$, which follows from $X_1(\mu_1)=Z_1+\mu_1\mid |Z_1+\mu_1|\le\kappa
\Rightarrow \kappa$ as $\mu_1\to\infty$. Repeating this argument coordinate by coordinate gives
\[
S_t(\mu)
\le
\max_{\varepsilon\in\{0,\infty\}^k}
S_t(\varepsilon).
\]
Therefore the supremum over \(\mu\in\mathbf R_+^k\) is equal to the maximum over the compactified
vertex set \(\{0,\infty\}^k\). By symmetry, the value depends only on the number
\(a\) of coordinates equal to \(+\infty\). Hence
\[
\sup_{\mu\in\mathbf R_+^k}S_t(\mu)
=
\max_{a=0,\ldots,k}S_t(\underbrace{\infty,\ldots,\infty}_{a},0,\ldots,0).
\]
It remains to compute these vertex values. If \(a\) coordinates are sent to
\(+\infty\), then those coordinates converge to \(+\kappa\). The remaining
\(k-a\) coordinates are i.i.d. \(N(0,1)\) variables truncated to
\([-\kappa,\kappa]\). Conditional on their signs, their squared magnitudes are
i.i.d. with CDF \(F_\kappa\), and each sign is positive or negative with
probability \(1/2\). If exactly \(r\) of the remaining \(k-a\) coordinates are
negative, then the positive and negative squared sums are
\[
a\kappa^2+\sum_{i=1}^{k-a-r}U_i',
\qquad
\sum_{i=1}^{r}U_i.
\]
Thus the corresponding CDF is exactly \(F^{(a)}_{k,\kappa}\). Therefore
\[
\inf_{\mu\in\mathbf R_+^k}L^{s,0}(t,\mu,I_k)
=
1-\sup_{\mu\in\mathbf R_+^k}S_t(\mu)
=
\min_{a=0,\ldots,k}F^{(a)}_{k,\kappa}(t).
\]
Finally, the conclusion follows by applying sign symmetry $L^{s,0}(t,\mu,I_k)
=
L^{s,0}(t,-\mu,I_k)$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:power-comparison}}
For simplicity let $\gamma=\alpha-\tau$ and write $$c^{\ell}_{L}=c^{\ell}(1-\alpha, I_k).$$ By Lemmas \ref{alem:consistency-sample-variance}-\ref{alem:uniformCLT} and Slutsky's theorem, we have
\[
\sqrt{n}S^{-1}_n \bar{W}_n
\Rightarrow
Z+\theta^{A,B,h,r},
\qquad Z=(Z_1,\cdots,Z_k)\sim N(0_k,I_k).
\]
Therefore the asymptotic power of each test is obtained by evaluating the corresponding Gaussian rejection probability at \(Z+\theta^{A,B,h,r}\). It is enough to compare the limiting Gaussian rejection probabilities.
We first prove (i). Suppose \(k=2\). The screening threshold in the conditional test is
$\kappa_\tau:=\kappa_\tau(I_k)= \Phi^{-1}( (1-\tau)^{\frac1k})$ given in \eqref{def:kappa-screening-threshold}. Since $\sqrt{1-\tau}>1-\tau > 1-\alpha$, this threshold is larger than the LF critical value with $\ell=m$:
\[
\kappa_\tau=\Phi^{-1}(\sqrt{1-\tau})> \Phi^{-1}(1-\alpha)=c^m(1-\alpha, I_k).
\]
For $k=2$, if $|\mathcal I_0|=1$, the conditional critical value used on the positive, negative, and origin branches coincide:
\begin{align}\label{eq:power-comparison-k2}
c^{m}_1=c^{m,-}_{\mathcal I_0}(1-\gamma,I_k)=c^{m,+}_{\mathcal I_0}(1-\gamma,I_k)=c^{m,0}(1-\gamma, I_k) .
\end{align}
Moreover, they are equal to
\[
c_1^m
=
-\Phi^{-1}(
\Phi(\kappa_\tau)
-
(2\Phi(\kappa_\tau)-1)(1-\gamma))
\overset{(a)}{=} \Phi^{-1}(\Phi(\kappa_\tau)-(2\Phi(\kappa_\tau)-1)\gamma),\]
where (a) follows from $-\Phi^{-1}(u)=\Phi^{-1}(1-u)$ for any $u\in(0,1)$. This implies
\begin{align*}
\Phi(c_1^m)= \Phi(\kappa_\tau)-(2\Phi(\kappa_\tau)-1)\gamma
\overset{(a)}{>} (1-2\gamma) (1-\tau)+\gamma
\overset{(b)}{>}1-\alpha=\Phi(c_L^m)
\end{align*}
where (a) holds because $\Phi(\kappa_\tau) = \sqrt{1-\tau}>1-\tau$ and (b) holds because $(1-2\gamma) (1-\tau)+\gamma=1-\alpha+2\tau(\alpha-\tau)$ and $\alpha>\tau>0$, which further yields $c_1^m>c_L^m. $
We now compare rejection regions. If the conditional test rejects immediately because one coordinate is screened positive and the other is screened negative,
then one coordinate exceeds \(\kappa_\tau\) and the other is below \(-\kappa_\tau\). Since \(\kappa_\tau>c_L^m\), the LF max test also rejects.
If the conditional test rejects on the positive or negative case and so $|\mathcal I_0|=1$, then one
coordinate exceeds \(\kappa_\tau\) in absolute value with one sign, and the unscreened coordinate exceeds \(c_1^m\) in the opposite direction by \eqref{eq:power-comparison-k2}. Since \(\kappa_\tau>c_L^m\) and \(c_1^m>c_L^m\), the LF max test also rejects.
Finally, if the conditional test rejects on the origin case, then the two coordinates must have opposite signs and both exceed \(c_1^m\) in absolute
value by \eqref{eq:power-comparison-k2}. Since \(c_1^m>c_L^m\), the LF max test again rejects. Hence the conclusion holds for $\ell=m$.
For \(\ell=s\), the same argument applies after squaring the thresholds. Specifically, for $\alpha<1/2$, the LF critical value satisfies $c_L^s=(c_L^m)^2$. Likewise, because $\gamma=\alpha-\tau<1/2,$ the one-dimensional conditional critical value is $c_1^s=(c_1^m)^2$. Moreover,
$$c^{s,0}=c_1^s.$$
To see this, note that, by the definition in \eqref{def:F-origin-conditional}, for $t\geq 0,$
\begin{align*}
F^{(2)}_{2,\kappa_\tau} (t) &= 1\\
F^{(1)}_{2,\kappa_\tau} (t) &= \sum^1_{r=0}{1\choose r} 2^{-1}P\{\min\{\kappa_\tau^2 + \sum^{1-r}_{i=1}U_i', \sum^{r}_{i=1}U_i\}\leq t\}
=\tfrac12 +\tfrac12 P\{U_i\leq t\}
=\tfrac12 +\tfrac12F_{\kappa_\tau}(t)\\
F^{(0)}_{2,\kappa_\tau} (t) &= \sum^2_{r=0}{2\choose r} 2^{-2}P\{\min\{\ \sum^{2-r}_{i=1}U_i', \sum^{r}_{i=1}U_i\}\leq t\}=\tfrac14+\tfrac12 P\{\min\{U'_i,U_i\}\leq t\}+\tfrac14 \\
&=\tfrac12 + F_{\kappa_\tau}(t) - \tfrac12F_{\kappa_\tau}(t)^2
\end{align*}
where $U_i', U_i\sim i.i.d. ~F_{\kappa_\tau}.$ Since $F^{(0)}_{2,\kappa_\tau} (t) - F^{(1)}_{2,\kappa_\tau} (t) = \tfrac12 F_{\kappa_\tau}(t)(1-F_{\kappa_\tau}(t))\geq0$ and $F^{(1)}_{2,\kappa_\tau} (t) \leq 1$, it holds that $\min_{a=0,1,2}F^{(a)}_{2,\kappa_\tau} (t)= F^{(1)}_{2,\kappa_\tau}(t)=\tfrac12 + \tfrac12 F_{\kappa_\tau}(t)$. Since this is the CDF where $c^{s}_1$ is computed from, we have $c^{s,0}=c_1^s$.
Then, because \(c_1^m>c_L^m\), we have \(c_1^s>c_L^s\). Therefore the same case-by-case inclusion of rejection regions holds for \(\ell=s\). Finally, when \(\Omega=I_2\), the quasi-likelihood ratio statistic coincides with the sum-of-squares statistic: $T^q(x,I_2)=T^s(x).$ Thus the same rejection-region inclusion proves the claim for \(\ell=q\). This proves (i).
We next prove (ii). Let \(k\ge3\) and \(h=1\). We first set \(B=0\) and let \(A\to\infty\). For the LF test, the first coordinate diverges to \(+\infty\). Hence the LF
test rejects if and only if the remaining \(k-1\) coordinates generate enough evidence in the negative direction. At \(B=0\), this is exactly the least
favorable configuration as shown in Theorem 1 by \cite{PiantadosiGail1993} for $\ell=m$ and Theorem 1 by \cite{GailSimon1985Biometrics} for $\ell=s,q$. Therefore, for each \(\ell\in\{m,s,q\}\), $\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\phi_n^\ell]
=
\alpha.$
We now compute the corresponding limiting rejection probability of the
conditional test at \(B=0\). Since the first coordinate diverges to \(+\infty\),
it is screened positive with probability tending to one. The remaining
\(k-1\) coordinates are independent standard normal variables. The conditional test rejects immediately if at least one of the remaining \(k-1\) coordinates is screened negative, whose probability is
\[
P\{ \exists Z_j \text{ s.t. }Z_j<-\kappa_\tau \text{ for }j=2,..,k\} = 1-P\{Z_j >-\kappa_\tau ~~\forall j=2,..,k\}=1-\Phi(\kappa_\tau)^{k-1}.
\]
If no remaining coordinate is screened negative, and at least one remaining coordinate is unscreened, then $I_{\textrm{pos}}=1$. The probability of this event is
\begin{align*}
&P\{\forall j\geq 2 ~~ Z_j\geq-\kappa_\tau,~~ \exists j\geq2~~ Z_j\leq \kappa_\tau\}
= P\{\forall j\geq2~~ Z_j\geq-\kappa_\tau\} -P\{\forall j\geq2 ~~Z_j> \kappa_\tau\}\\
&=\Phi(\kappa_\tau)^{k-1} - (1-\Phi(\kappa_\tau))^{k-1}.
\end{align*}
The conditional subtest $\psi^{\ell,+}_n$ in \eqref{def:conditional-test-pos} has rejection probability \(\gamma\) at the least favorable value $0_k$ by Proposition \ref{prop:conditional-lf-pos-branch}. Therefore, at \(B=0\),
\begin{align*}
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\psi_n^\ell]
&\leq
1-\Phi(\kappa_\tau)^{k-1}
+
(\alpha-\tau)
\{
\Phi(\kappa_\tau)^{k-1} - (1-\Phi(\kappa_\tau))^{k-1}
\}\\
&<
1- (1-\alpha + \tau) \Phi(\kappa_\tau)^{k-1}
<
1- (1-\alpha + \tau) \Phi(\kappa_\tau)^{k}\\
&\overset{(a)}{=}1- (1-\alpha + \tau)(1-\tau) = \alpha-\tau(\alpha-\tau)\\
&<\alpha=\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\phi_n^\ell]
\end{align*}
where (a) holds because $\kappa_\tau= \Phi^{-1}( (1-\tau)^{\frac1k})$ so $\Phi(\kappa_\tau)^k=1-\tau.$
It remains to extend the strict inequality to small \(B>0\). For fixed \(r\in\{1,\ldots,k-1\}\), both limiting Gaussian rejection probabilities are continuous in \(B\). Since the inequality is strict at \(B=0\), there exists
\(B_0^\ell(r)>0\) such that the same inequality holds for every \(B\in(0,B_0^\ell(r)]\). Take $B_0^\ell:=\min_{1\le r\le k-1} B_0^\ell(r)>0.$ Then the conclusion follows.
We prove (iii). As above, it is enough to compare the limiting Gaussian rejection probabilities after taking \(A\to\infty\). Since the first \(h\) coordinates diverge to \(+\infty\), they are screened positive with probability tending to one. Hence the conditional test immediately rejects whenever at least one of the remaining coordinates is screened negative. If no remaining coordinate is screened negative, then either all remaining coordinates are screened positive, in which case the conditional test does not reject, or at least one remaining coordinate is unscreened, in which case \(I_{\mathrm{pos}}=1\). The number of unscreened coordinates is at most \(k-h\). Under \(\Omega=I_k\), the conditional statistics used when \(I_{\mathrm{pos}}=1\) can be coupled across values of \(|\mathcal I_0|\) by appending independent truncated coordinates. The maximum statistic for \(\ell=m\) and the nonnegative sum statistic for \(\ell=s,q\) are then weakly increasing, so \(c_d^\ell\) is weakly increasing in \(d\).
For \(\ell=m\), let \(Z_1,\ldots,Z_{k-h}\) be independent standard normal random variables. After taking \(A\to\infty\), the event
$
\{
\min\{
\min_{1\le j\le r}(Z_j-B),
\min_{r<j\le k-h}Z_j
\}
<-c_{k-h}^m
\}
$
implies rejection by the conditional max test. Therefore,
\begin{align*}
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\psi_n^m]
&\ge
P\left\{
\min\left\{
\min_{1\le j\le r}(Z_j-B),
\min_{r<j\le k-h}Z_j
\right\}
<-c_{k-h}^m
\right\}\\
&=
1-\Phi(c_{k-h}^m-B)^r
\Phi(c_{k-h}^m)^{k-h-r}.
\end{align*}
On the other hand, the limiting rejection probability of the least
favorable max test is exactly
\[
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\phi_n^m]
=
1-
\Phi\bigl(c^m_L-B\bigr)^r
\Phi\bigl(c^m_L\bigr)^{k-h-r}.
\]
Under \eqref{def:cv-comparison} that $c^m_{k-h}<c^m_L$, we have
\[
\begin{aligned}
&\Phi(c_{k-h}^m-B)^r
\Phi(c_{k-h}^m)^{k-h-r}<
\Phi\bigl(c^m_L-B\bigr)^r
\Phi\bigl(c^m_L\bigr)^{k-h-r},
\end{aligned}
\]
because \(\Phi\) is strictly increasing, \(r\ge1\), and
\(k-h-r\ge0\). Hence
\[
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\psi_n^m]
>
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\phi_n^m].
\]
We next consider \(\ell=s\). By assumption, \(r=k-h\), so all the remaining
coordinates have local drift \(-B\). The event
\(
\{
\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\}
>
c_{k-h}^s
\}
\)
implies rejection by the conditional test. If some remaining coordinate is
screened negative, rejection is immediate. Otherwise, the displayed event
implies that at least one remaining coordinate is unscreened, so \(I_{\mathrm{pos}}=1\). Coordinates screened positive make
zero contribution to the sum, and every negative coordinate is
unscreened. Thus, when \(I_{\mathrm{pos}}=1\), the statistic equals $\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\},$ while the relevant critical value is no larger than \(c_{k-h}^s\). Therefore,
\[
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\psi_n^s]
\ge
P\{
\textstyle\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\}
>
c_{k-h}^s
\}.
\]
The limiting rejection probability of the least favorable sum-of-squares
test is exactly
\[
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\phi_n^s]
=
P\{
\textstyle\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\}
>
c^s(1-\alpha,I_k)
\}.
\]
The random variable
\(
\textstyle\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\}
\)
has an atom at zero and a strictly positive density on \((0,\infty)\).
Moreover, \(c^s_L>0\), because the defining least favorable
distribution has probability \(2^{-(k-1)}<1-\alpha\) at zero. Hence
\eqref{def:cv-comparison} implies
\[
\begin{aligned}
&P\{
\textstyle\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\}
>
c_{k-h}^s
\}>
P\{
\textstyle\sum_{j=1}^{k-h}
(Z_j-B)^2I\{Z_j\le B\}
>
c^s_L
\}.
\end{aligned}
\]
It follows that
\[
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\psi_n^s]
>
\lim_{A\to\infty}\lim_{n\to\infty}
\mathbb E_{P_n}[\phi_n^s].
\]
When \(\Omega=I_k\), the quasi-likelihood ratio statistic coincides with the corresponding sum-of-squares distance to the null cone, both for the least favorable test and in the positive case of the conditional test. Therefore, the same argument applies. This proves (iii).
\section{Supporting technical results}
\subsection{Preliminary Results}
We begin by presenting three existing lemmas that are used repeatedly throughout the appendix.
\begin{lemma}[Lemma S.6.1. in \cite{RomanoShaikh2012AoS}]\label{alem:consistency-sample-variance}
Consider any sequence $\left\{P_n \in {\mathbf{P}}: n \geq 1\right\}$, where ${\mathbf{P}}$ satisfies \eqref{def:uniform-integrability}. Let $X_{i}, i=$ $1, \ldots, n$ be an i.i.d. sequence of real-valued random variables with distribution $P_n$. Then,
$$
\frac{S_n^2}{\sigma^2\left(P_n\right)} \stackrel{P_n}{\rightarrow} 1 .
$$
\end{lemma}
\begin{lemma}[Lemma S.7.1. in \cite{RomanoShaikh2012AoS}]\label{alem:consistency-sample-corr}
Consider any sequence $\left\{P_n \in {\mathbf{P}}: n \geq 1\right\}$, where ${\mathbf{P}}$ satisfies \eqref{def:uniform-integrability}. Let $X_{ i}, i=1, \ldots, n$ be an i.i.d. sequence of random variables with distribution $P_n$. Then,
$$
\|\Omega(\hat{P}_n)-\Omega(P_n)\| \stackrel{P_n}{\rightarrow} 0,
$$
where the norm $\|\cdot\|$ is the component-wise maximum of the absolute value of all elements.
\end{lemma}
\begin{lemma}[Lemma 3.1. in \cite{RomanoShaikh2008JSPI}]\label{alem:uniformCLT}
Let $\mathbf{P}$ be a set of distributions on $\mathbf{R}^k$ such that the marginal distributions satisfy \eqref{def:uniform-integrability}. Let $X_i=$ $\left(X_{1, i}, \ldots, X_{k, i}\right), i=1, \ldots, n$ be an i.i.d. sequence of random variables with distribution $P \in \mathbf{P}$. Denote by $\Phi_V(\cdot)$ the probability distribution of a multivariate normal random variable with mean $0_k$ and variance $V$ and by $\Sigma(P)$ the variance of $P$. Then,
$$
\sup _{P \in \mathbf{P}} \sup _{S \in \mathcal{S}} \mid P\left\{\left(\sqrt{n}\left(\bar{X}_{1, n}-\mu_1(P), \ldots, \sqrt{n}\left(\bar{X}_{k, n}-\mu_k(P)\right) \in S\right\}-\Phi_{\Sigma(P)}(S) \mid \rightarrow 0,\right.\right.
$$
where $\mathcal{S}=\left\{S \subseteq \mathbf{R}^k: S\right.$ convex and $\Phi_V(\partial S)=0$ for all p.s.d. $V$ s.t. $V_{i, i}=1$ for $\left.1 \leqslant i \leqslant k\right\}$.
\end{lemma}
\subsection{Lemmas for the least favorable test}
\subsubsection{Lemmas for the least favorable critical values}
\begin{lemma}[Existence]\label{lem:parametric-cv-existence}
Consider the parametric bootstrap critical value in
\eqref{def:LF-cv-normal} and fix $\alpha\in(0,1)$.
For $\ell\in\{m,s\}$ and any positive semidefinitecorrelation matrix
$\theta_2\in\bar{\mathbf O}$, $c^\ell(1-\alpha,\theta_2)$ is well-defined.
Moreover,
\begin{align}\label{eq:cv-upperbound1}
c^m(1-\alpha,\theta_2)
\le
\Phi^{-1}\left(1-\frac{\alpha}{k}\right),\quad
c^s(1-\alpha,\theta_2)
\le
k\left\{\Phi^{-1}\left(1-\frac{\alpha}{2k}\right)\right\}^2.
\end{align}
For $\ell=q$, the same conclusion holds for any $\theta_2\in\mathbf O$. In particular,
\begin{align}\label{eq:cv-upperbound2}
c^q(1-\alpha,\theta_2)
\le
\chi^2_{k,1-\alpha},
\end{align}
where $\chi^2_{k,1-\alpha}$ denotes the $(1-\alpha)$-quantile of the
chi-squared distribution with $k$ degrees of freedom.
\end{lemma}
\begin{proof}
Fix $\theta_1\in\mathbf R^k_-$ and
$\theta_2\in\bar{\mathbf O}$. Since
$T^\ell(\theta_2^{1/2}Z+\theta_1,\theta_2)$ is real-valued for
$\ell \in \{m,s\}$, $J^\ell(\cdot,\theta_1,\theta_2)$ in \eqref{def:J} is a proper CDF and
$(J^\ell)^{-1}(1-\alpha,\theta_1,\theta_2)=\inf\{x\in\mathbf{R}: P\{T^{\ell}(\theta_2^{1/2}Z +\theta_1, \theta_2)\leq x\}\geq 1-\alpha\}$ is finite. Moreover, the set over
which the supremum is taken in \eqref{def:LF-cv-normal} is non-empty since
$0_k\in\mathbf R^k_+\cup\mathbf R^k_-$. It remains to show that the quantiles are uniformly bounded from above.
Consider $\ell=m$. For any
$x\in\mathbf R$,
\begin{align*}
1-J^m(x,\theta_1,\theta_2)
&=
P\{T^m(\theta_2^{1/2}Z+\theta_1)>x\}
\overset{(a)}{\le}
P\left\{\max_{1\le j\le k}(\theta_2^{1/2}Z+\theta_1)_j>x\right\}
\\
&\overset{(b)}{\le}
P\left\{\max_{1\le j\le k}(\theta_2^{1/2}Z)_j>x\right\}
\overset{(c)}{\le}
\sum_{j=1}^k P\{(\theta_2^{1/2}Z)_j>x\}
=
k\{1-\Phi(x)\}.
\end{align*}
where (a) holds because
$T^m(y)\le \max_{1\le j\le k}y_j$ for any $y\in\mathbf{R}^k$; (b) holds because
$\theta_1\in\mathbf R_-^k$, so
$(\theta_2^{1/2}Z+\theta_1)_j\le(\theta_2^{1/2}Z)_j$ for all $j$; (c) is the union bound. The last equality holds because
$Var((\theta_2^{1/2}Z)_j)=(\theta_2)_{jj}=1$, and hence
$(\theta_2^{1/2}Z)_j\sim N(0,1)$. Taking $x=\Phi^{-1}(1-\alpha/k)$ gives $J^m(\Phi^{-1}(1-\frac{\alpha}{k}),
\theta_1,\theta_2)
\ge 1-\alpha.$
Therefore, for all $\theta_1\in\mathbf{R}^k_-$, it holds $(J^m)^{-1}(1-\alpha,\theta_1,\theta_2)
\le
\Phi^{-1}\left(1-\frac{\alpha}{k}\right)$ and the upper bound of $c^m(1-\alpha,\theta_2)$ in \eqref{eq:cv-upperbound1} holds.
Next, let $\ell=s$. For any $x\ge0$,
\begin{align*}
1-J^s(x,\theta_1,\theta_2)
&\overset{(a)}{\le}
P\left\{
\sum_{j=1}^k
(\theta_2^{1/2}Z+\theta_1)_j^2
I\{(\theta_2^{1/2}Z+\theta_1)_j\ge0\}
>x
\right\} \\
&\overset{(b)}{\le}
P\left\{
\sum_{j=1}^k
\left((\theta_2^{1/2}Z)_j\right)^2
>x
\right\}
\overset{(c)}{\le}
P\left\{
\max_{1\le j\le k}|(\theta_2^{1/2}Z)_j|>\sqrt{x/k}
\right\} \\
&\overset{(d)}{\le}
\sum_{j=1}^k
P\left\{|(\theta_2^{1/2}Z)_j|>\sqrt{x/k}\right\}
\overset{(e)}{=}
2k\left\{1-\Phi\left(\sqrt{x/k}\right)\right\}
\end{align*}
where (a) holds by the definition of $T^s$; (b) holds because $\theta_1\in\mathbf R_-^k$: whenever
$(\theta_2^{1/2}Z+\theta_1)_j\ge0$, we have $0\le
(\theta_2^{1/2}Z+\theta_1)_j
\le
(\theta_2^{1/2}Z)_j$; (c) holds trivially; (d) holds by the union
bound; finally, (e) again uses
$(\theta_2^{1/2}Z)_j\sim N(0,1)$. Taking $x=
k\{\Phi^{-1}(1-\frac{\alpha}{2k})\}^2$ gives $J^s(
k\{\Phi^{-1}(1-\frac{\alpha}{2k})\}^2,
\theta_1,\theta_2
)
\ge 1-\alpha.$ Therefore, for all $\theta_1\in\mathbf{R}^k_-$, $(J^s)^{-1}(1-\alpha,\theta_1,\theta_2)
\le
k\left\{\Phi^{-1}\left(1-\frac{\alpha}{2k}\right)\right\}^2$ holds, so the upper bound of $c^s(1-\alpha,\theta_2)$ in \eqref{eq:cv-upperbound1} holds.
Finally consider $\ell=q$ and suppose $\theta_2\in\mathbf O$, so that
$\theta_2^{-1}$ exists. For any
$\theta_1\in\mathbf R^k_-$, $\mu=\theta_1$ is feasible in the
definition of $T^q$. Hence
\begin{align*}
T^q(\theta_2^{1/2}Z+\theta_1,\theta_2)
&=
\inf_{\mu\in\mathbf R^k_+\cup\mathbf R^k_-}
(\theta_2^{1/2}Z+\theta_1-\mu)'
\theta_2^{-1}
(\theta_2^{1/2}Z+\theta_1-\mu)
\le
Z'Z.
\end{align*}
where the inequality holds because $\mu=\theta_1$ belongs to
$\mathbf R^k_+\cup\mathbf R^k_-$. Therefore, for any $x\in\mathbf R$,
\[
1-J^q(x,\theta_1,\theta_2)
\le
P\{Z'Z>x\}
=
P\{\chi^2_k>x\}.
\]
Taking $x=\chi^2_{k,1-\alpha}$ gives $(J^q)^{-1}(1-\alpha,\theta_1,\theta_2)
\le
\chi^2_{k,1-\alpha}.$ Taking the supremum over
$\theta_1\in\mathbf R^k_-$ yields \eqref{eq:cv-upperbound2}.
\end{proof}
For \(\theta_1\in[0,\infty]^k\), define
\[
\mathbf I(\theta_1):=\{j\in[k]:\theta_{1,j}=+\infty\},
\qquad
\mathbf I^c(\theta_1):=[k]\setminus \mathbf I(\theta_1).
\]
For a vector \(a\), write \(a_I\) for the subvector indexed by \(I\), and for a matrix
\(A\), write \(A_{I\times I}\) for the corresponding principal submatrix. Furthermore, for brevity, we use $\left(J^{\ell}\right)^{-1}$ and $J^{\ell,-1}$ interchangeably.
\begin{lemma}\label{lem:argmax-critical-value-all}
Fix \(\alpha\in(0,\tfrac12)\). For \(\ell\in\{m,s\}\), define the extended inverse
\(\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)\) where
\((\theta_1, \theta_2)\in[0,\infty]^k\times\bar{\mathbf O}\) as follows.
For \(\ell=m\),
\[
\bar J^{m,-1}(1-\alpha,\theta_1,\theta_2)
:=
\begin{cases}
J^{m,-1}(1-\alpha,\theta_1,\theta_2),
& \mathbf I(\theta_1)=\varnothing,\\
\inf\{x:
P\{\max_{j\in\mathbf I^c(\theta_1)}(-Z_j-\theta_{1,j})\le x\}\ge1-\alpha\},
& \varnothing\subsetneq\mathbf I(\theta_1)\subsetneq[k],\\
-\infty,
& \mathbf I(\theta_1)=[k],
\end{cases}
\]
where \(Z=(Z_1, ..., Z_k)\sim N(0_k,\theta_2)\).
For \(\ell=s\), let $X^s:=\sum_{j\in\mathbf I^c(\theta_1)}(Z_j+\theta_{1,j})^2
I\{Z_j+\theta_{1,j}\le0\}$ and
\[
\bar J^{s,-1}(1-\alpha,\theta_1,\theta_2)
:=
\begin{cases}
J^{s,-1}(1-\alpha,\theta_1,\theta_2),
& \mathbf I(\theta_1)=\varnothing,\\
\inf\{x:
P\{X^s\le x\}\ge1-\alpha\},
& \varnothing\subsetneq\mathbf I(\theta_1)\subsetneq[k],\\
0,
& \mathbf I(\theta_1)=[k].
\end{cases}
\]
For \(\ell=q\), if \(\theta_2\succ0\), define $Y:=(Z+\theta_1)_{\mathbf I^c(\theta_1)}
\sim
N(\theta_{1,\mathbf I^c(\theta_1)},\theta_{2,\mathbf I^c(\theta_1)\times\mathbf I^c(\theta_1)})$ and $X^q:=\inf_{\nu\in\mathbf R_+^{|\mathbf I^c(\theta_1)|}}
(Y-\nu)' (\theta_{2,\mathbf I^c(\theta_1)\times\mathbf I^c(\theta_1)})^{-1}(Y-\nu)$ and
\[
\bar J^{q,-1}(1-\alpha,\theta_1,\theta_2)
:=
\begin{cases}
J^{q,-1}(1-\alpha,\theta_1,\theta_2),
& \mathbf I(\theta_1)=\varnothing,\\
\inf\{x:
P\{ X^q\le x \}
\ge1-\alpha\},
& \varnothing\subsetneq\mathbf I(\theta_1)\subsetneq[k],\\
0,
& \mathbf I(\theta_1)=[k].
\end{cases}
\]
Then, \((\theta_1,\theta_2)\mapsto\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)\) is continuous on $[0,\infty]^k\times \bar{\mathbf{O}}$ for $\ell=m,s$ and on $[0,\infty]^k\times \mathbf{O}$ for $\ell=q$.
Furthermore, for each \(\theta_2\in\bar{\mathbf O}\), define
\[
\mathbf S^\ell(\theta_2)
:=
\arg\max_{\theta_1\in[0,\infty]^k}
\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2),
\qquad \ell\in\{m,s\}.
\]
Then, \(\mathbf S^\ell(\theta_2)\) is nonempty and compact, and satisfies
\[
c^\ell(1-\alpha,\theta_2)
=
\max_{\theta_1\in[0,\infty]^k}
\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)
=
\sup_{\theta_1\in[0,\infty)^k}
J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)
\]
for \(\ell\in\{m,s\}\). The same conclusion holds for \(\ell=q\) for
\(\theta_2\in\mathbf O\).
\end{lemma}
\begin{proof}
Fix \(\ell\in\{m,s\}\). For \(\ell=q\), restrict attention to a set of correlation matrices whose smallest eigenvalue is bounded away from zero. Consider any convergent sequence $(\theta_{1,n},\theta_{2,n})\to (\theta_{1,\infty},\theta_{2,\infty})$ in $[0,\infty]^k\times\bar{\mathbf O}.$ For \(\ell=q\), assume \(\theta_{2,\infty}\succ0\) then we can find a sequence \(\theta_{2,n}\in\mathbf{O}\) such that $\theta_{2,n}\to \theta_{2,\infty}$.
If \(\mathbf I(\theta_{1,\infty})=\varnothing\), then all coordinates of \(\theta_{1,n}\) have finite limits. Since the the principal square-root map is continuous, $\theta_{2,n}^{1/2}Z+\theta_{1,n}
\Rightarrow
\theta_{2,\infty}^{1/2}Z+\theta_{1,\infty}.$ By continuity of \(T^m\), \(T^s\), and \(T^q\) on their domains, we obtain $T^\ell(\theta_{2,n}^{1/2}Z+\theta_{1,n},\theta_{2,n})
\Rightarrow
T^\ell(\theta_{2,\infty}^{1/2}Z+\theta_{1,\infty},\theta_{2,\infty}),$ where the second argument is ignored for \(\ell=m,s\).
If \(\varnothing\subsetneq \mathbf I(\theta_{1,\infty})\subsetneq[k]\), then the coordinates indexed by \(\mathbf I(\theta_{1,\infty})\) diverge to \(+\infty\), while the coordinates indexed by \(\mathbf I(\theta_{1,\infty})^c\) have finite limits.
Therefore,
\[
T^m(\theta_{2,n}^{1/2}Z+\theta_{1,n})
\Rightarrow
\textstyle\max_{j\in \mathbf I(\theta_{1,\infty})^c}\{-(\theta_{2,\infty}^{1/2}Z)_j-\theta_{1,\infty,j}\},
\]
\[
T^s(\theta_{2,n}^{1/2}Z+\theta_{1,n})
\Rightarrow
\textstyle\sum_{j\in \mathbf I(\theta_{1,\infty})^c}
((\theta_{2,\infty}^{1/2}Z)_j+\theta_{1,\infty,j})^2
I\{(\theta_{2,\infty}^{1/2}Z)_j+\theta_{1,\infty,j}\le0\}.
\]
For \(\ell=q\), the distance to \(\mathbf R_-^k\) diverges, while the distance to
\(\mathbf R_+^k\) reduces to the finite coordinates. Thus
\[
T^q(\theta_{2,n}^{1/2}Z+\theta_{1,n},\theta_{2,n})
\Rightarrow
\textstyle\inf\{
(Y-\nu)'(\theta_{2,\infty,\mathbf I(\theta_{1,\infty})^c\times \mathbf I(\theta_{1,\infty})^c})^{-1}(Y-\nu):{\nu\in\mathbf R_+^{|\mathbf I(\theta_{1,\infty})^c|}}\},
\]
where $Y=(\theta_{2,\infty}^{1/2}Z+\theta_{1,\infty})_{\mathbf I(\theta_{1,\infty})^c}.$ If \(\mathbf I(\theta_{1,\infty})=[k]\), then all coordinates diverge to \(+\infty\). Hence
\(T^m(\theta_{2,n}^{1/2}Z+\theta_{1,n})\overset{p}{\to}-\infty\), $T^s(\theta_{2,n}^{1/2}Z+\theta_{1,n})\overset{p}{\to}0$, and $T^q(\theta_{2,n}^{1/2}Z+\theta_{1,n},\theta_{2,n})\overset{p}{\to}0.$ These weak convergence statements are exactly the distributions $\bar J(\cdot, )$ used in the definition of
\(\bar J^{\ell,-1}\).
For \(\ell=m\), the limiting distribution is continuous and strictly increasing wherever its CDF takes values in \((0,1)\) by Lemma \ref{lem:strictly_increasing_dist_T}. Therefore, Lemma 11.2.1 of \cite{LehmannRomano2005testing} implies
\begin{align}\label{eq:quantile-convergence}
\bar J^{\ell,-1}(1-\alpha,\theta_{1,n},\theta_{2,n})
\to
\bar J^{\ell,-1}(1-\alpha,\theta_{1,\infty},\theta_{2,\infty}).
\end{align}
For $\ell\in\{s,q\}$, there are two possible cases. If $\bar J^{\ell,-1}(1-\alpha,\theta_{1,\infty},\theta_{2,\infty})>0$, then since $\bar J(\cdot, \theta_{1,\infty},\theta_{2,\infty})$ is continuous and strictly increasing at its $(1-\alpha)$ quantile by Lemma \ref{lem:strictly_increasing_dist_T}, Lemma 11.2.1(i) of \cite{LehmannRomano2005testing} gives \eqref{eq:quantile-convergence} as well. Suppose instead that $\bar J^{\ell,-1}(1-\alpha,\theta_{1,\infty},\theta_{2,\infty})=0.$ Because the statistics are nonnegative, $\bar J^{\ell,-1}(1-\alpha,\theta_{1,n},\theta_{2,n})\geq0$. Choose a continuity point
$x_\varepsilon\in(0,\varepsilon)$. Since
$\bar J^{\ell}(0,\theta_{1,\infty},\theta_{2,\infty})\geq1-\alpha$ and
$\bar J^{\ell}(x,\theta_{1,\infty},\theta_{2,\infty})>\bar J^{\ell}(0,\theta_{1,\infty},\theta_{2,\infty})$ for every $x>0$ whenever $\bar J^{\ell}(0,\theta_{1,\infty},\theta_{2,\infty})<1$, we have $\bar J^{\ell}(x_\varepsilon,\theta_{1,\infty},\theta_{2,\infty})>1-\alpha.$ Weak convergence therefore gives
$\bar J^{\ell}(x_\varepsilon,\theta_{1,n},\theta_{2,n}) (x_\varepsilon)>1-\alpha$ eventually, and hence $0\leq \bar J^{\ell,-1}(1-\alpha,\theta_{1,n},\theta_{2,n}) \leq x_\varepsilon<\varepsilon.$ Thus, $J^{\ell,-1}(1-\alpha,\theta_{1,n},\theta_{2,n}) (x_\varepsilon)\to 0=J^{\ell,-1}(1-\alpha,\theta_{1,\infty},\theta_{2,\infty})$. Therefore, \((\theta_1,\theta_2)\mapsto\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)\) is continuous on $[0,\infty]^k\times \bar{\mathbf{O}}$ for $\ell=m,s$ and on $[0,\infty]^k\times \mathbf{O}$ for $\ell=q$.
Now fix \(\theta_2 \in \bar{\mathbf{O}}\) for $\ell=m,s$ and \(\theta_2 \in {\mathbf{O}}\) for $\ell=q$. Since \([0,\infty]^k\) is compact and $\theta_1\mapsto\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)$ is continuous, the maximum is attained by the Weierstrass theorem. Hence
\(\mathbf S^\ell(\theta_2)\) is nonempty. Since it is the argmax of a continuous function on a compact set, it is closed and therefore compact. Finally, \([0,\infty)^k\) is dense in \([0,\infty]^k\), and \(\bar J^{\ell,-1}\) is a continuous extension of \(J^{\ell,-1}\). Therefore,
\[
\max_{\theta_1\in[0,\infty]^k}
\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)
=
\sup_{\theta_1\in[0,\infty)^k}
J^{\ell,-1}(1-\alpha,\theta_1,\theta_2).
\]
\end{proof}
\begin{lemma}[Continuity in correlation]\label{lem:continuity-critical-value-all}
Fix \(\alpha\in(0,\tfrac12)\). For \(\ell\in\{m,s\}\), the least favorable critical value
$c^\ell(1-\alpha,\theta_2)$ in \eqref{def:LF-cv-normal} is continuous on \(\bar{\mathbf O}\). For \(\ell=q\), the same conclusion holds on $\mathbf{O}$.
\end{lemma}
\begin{proof}
For \(\ell\in\{m,s\}\), Lemma \ref{lem:argmax-critical-value-all} gives $c^\ell(1-\alpha,\theta_2)
=
\max_{\theta_1\in[0,\infty]^k}
\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2).$ The feasible correspondence \([0,\infty]^k\) is constant, compact-valued and does not depend on \(\theta_2\). The objective $(\theta_1,\theta_2)\mapsto \bar{J}^{\ell,-1}(1-\alpha,\theta_1, \theta_2)$ is continuous on
\([0,\infty]^k\times\bar{\mathbf O}\). Therefore, Berge's maximum theorem implies that $\theta_2\mapsto c^\ell(1-\alpha,\theta_2)$ is continuous on \(\bar{\mathbf O}\). For \(\ell=q\), \((\theta_1,\theta_2)\mapsto\bar J^{\ell,-1}(1-\alpha,\theta_1,\theta_2)\) is continuous on $[0,\infty]^k\times\mathbf O.$ Hence, by Berge's theorem, $\theta_2\mapsto c^q(1-\alpha,\theta_2)$ is continuous on \(\mathbf O\).
\end{proof}
\subsubsection{Lemmas for Theorem \ref{thm:LFtest-uniform-validity}}
\begin{lemma}\label{lem:statistic-degenerate-case}
Consider a sequence $\{P_n\in \mathbf{P}: n \geq 1\}$ where $\mathbf{P}$ is a set of distributions on $\mathbf{R}^k$ satisfying \eqref{def:uniform-integrability}. Let $W_i$, $i=1,2,...,n$, be an i.i.d. sequence of random vectors with distribution $P_n$. Suppose
\begin{align*}
\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to -\infty \text{ for all }1\leq j\leq k \quad\text{or}\quad\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \infty \text{ for all }1\leq j\leq k.
\end{align*}
Then, $T^m_n \overset{P_n}{\to} -\infty,$ and $T^s_n\overset{P_n}{\to}0$. Furthermore, if the smallest eigenvalue of $\Omega(P)$ is bounded away from zero uniformly over $\mathbf{P}$, $\inf_{P \in \mathbf{P}} \lambda_{\min}(\Omega(P))>0$, $T^q_n\overset{P_n}{\to}0$.
\end{lemma}
\begin{proof}
For brevity of notation, consider
\begin{align*}
X_n:=\sqrt{n}S_n^{-1}\bar W_n
&=
S^{-1}_nD(P_n) \cdot D^{-1}(P_n) \sqrt{n}(\bar{W}_n-\mu(P_n))
+ S^{-1}_n\sqrt{n}\mu(P_n).
\end{align*}
The first term is stochastically bounded, i.e., $ S^{-1}_nD(P_n) \cdot D^{-1}(P_n) \sqrt{n}(\bar{W}_n-\mu(P_n)) =O_{P_n}(1)$ because $S^{-1}_nD(P_n)=I_k + o_{P_n}(1)$ by Lemma \ref{alem:consistency-sample-variance}, $D^{-1}(P_n) \sqrt{n}(\bar{W}_n-\mu(P_n)) = O_{P_n}(1)$ by Lemma \ref{alem:uniformCLT}, and by Slutsky's theorem. For the second term, by Lemma \ref{alem:consistency-sample-variance},
\begin{align*}
\tfrac{\sqrt{n}\mu_j(P_n)}{S_{j,n}} \overset{P_n}{\to} -\infty
\quad\text{if }
\tfrac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to -\infty,
\quad
\tfrac{\sqrt{n}\mu_j(P_n)}{S_{j,n}} \overset{P_n}{\to} \infty
\quad\text{if }
\tfrac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \infty
\end{align*}
for any $j=1,...,k.$ Combining these two terms gives
\begin{align}\label{eq:statistic-degenerate-case-eq2}
X_{j,n}
=
\tfrac{\sqrt{n}\bar{W}_{j,n}}{S_{j,n}}
\overset{P_n}{\to} -\infty
\quad\text{if }
\tfrac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to -\infty,
\quad
X_{j,n}
=
\tfrac{\sqrt{n}\bar{W}_{j,n}}{S_{j,n}}
\overset{P_n}{\to} \infty
\quad\text{if }
\tfrac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \infty
\end{align}
for any $j=1,...,k.$
Next, I show that $T^m_n$ diverges to $-\infty$ in probability. Suppose first that
$\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to -\infty$ for all $j=1,...,k.$
By \eqref{eq:statistic-degenerate-case-eq2}, for any $\varepsilon>0$ and $M<0$, there exists $N \in \mathbf{N}$ such that $\sup_{n\geq N} P_n\{X_{j,n} \geq M\} < \frac{\varepsilon}{k}.$ Then,
\begin{align*}
P_n\{T^m_n > M\}
&= P_n\{T^m_n > M, \quad\forall j\leq k~X_{j,n} < M\}
+ P_n\{T^m_n > M, \quad\exists j\leq k~X_{j,n} \geq M\}\\
&\leq
P_n\{T^m_n > M, \quad\forall j\leq k~X_{j,n} < M\}
+ \varepsilon\\
&\leq
P_n\{\textstyle\max_{1\leq j \leq k} X_{j,n}> M,
\quad\forall j\leq k~X_{j,n} < M\}
+ \varepsilon
=
\varepsilon,
\end{align*}
where the first inequality holds because $ P_n\{\exists j\leq k~X_{j,n} \geq M\}
\leq
\textstyle\sum^k_{j=1} P_n\{X_{j,n} \geq M\}
<\varepsilon,$
and the second inequality holds because $\{T^m_n>M\}\subset\{\max_{1\leq j\leq k}X_{j,n}>M\}$. As a result, it holds that
$T^m_n\overset{P_n}{\to}-\infty.$ A similar argument proves that
$T^m_n\overset{P_n}{\to}-\infty$ if
$\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \infty$ for all $j=1,...,k.$
The convergence $T^s_n\overset{P_n}{\to}0$ holds from the following: for any $\varepsilon>0$,
$$P_n\{|T^s_n|>\varepsilon\} \leq P_n\{|T^s_n|>0\}\leq 1-P_n\{T^s_n=0\}=1-P_n\{X_n\in\mathbf{R}^k_+\cup\mathbf{R}^k_-\}\to0$$
where the convergence holds by \eqref{eq:statistic-degenerate-case-eq2}. Similarly, the convergence of $T^q_n\overset{P_n}{\to}0$ holds from the following: for any $\varepsilon>0$,
\begin{align*}
P_n\{|T^q_n|>\varepsilon\}
&\overset{(a)}{=} P_n\{|T^q_n|>\varepsilon, \hat\Omega_n\succ0\} + o(1) \overset{(b)}{\le} P_n\{|T^q_n|>0, \hat\Omega_n\succ0\} + o(1)\\
&\overset{(c)}{\le} P_n\{X_n \notin \mathbf{R}^k_+\cup\mathbf{R}^k_-, \hat\Omega_n\succ0\} + o(1) \overset{(d)}{\to}0
\end{align*}
where (a) holds as $\hat\Omega_n$ is positive definite with probability approaching one under the assumption on positive minimum eigenvalue; (b) holds by monotonicity; (c) holds because $T^q_n=0$ if and only if $X_n\in \mathbf{R}^k_+\cup\mathbf{R}^k_-$; and finally (d) holds by \eqref{eq:statistic-degenerate-case-eq2}.
\end{proof}
\begin{lemma}\label{lem:conv-ip-to-Jn}
Consider a sequence $\{P_n\in \mathbf{P}: n \geq 1\}$ where $\mathbf{P}$ is a set of distributions on $\mathbf{R}^k$ satisfying \eqref{def:uniform-integrability}. Let $W_i$, $i=1,2,...,n$, be an i.i.d. sequence of random vectors with distribution $P_n$. Suppose for some non-empty set $I\subset \{1,2, ..., k\}$
\begin{align}\label{eq:conv-ip-to-Jn-condition1}
\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \delta_j \in(-\infty, 0] \text{ for } j\in I\text{ and }\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to -\infty \text{ for }j \not\in I
\end{align}
or
\begin{align}\label{eq:conv-ip-to-Jn-condition2}
\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \delta_j \in [0,\infty) \text{ for } j\in I\text{ and }\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \infty \text{ for }j \not\in I.
\end{align}
Then, for $\ell\in\{m,s\}$, $J^{\ell}$ in \eqref{def:J} and $J^{\ell}_n(x):=J^{\ell}_n(x,\sqrt{n}\mu(P_n),P_n)$ in \eqref{def:dist-test-stat} satisfy
\begin{align}\label{eq:conv-ip-to-Jn}
\sup_{x \in \mathbf{R}} | J^{\ell}(x, \sqrt{n}S^{-1}_n\mu(P_n), \Omega(\hat{P}_n)) - J^{\ell}_n(x)| \overset{P_n}{\to} 0 \text{ as }n\to\infty.
\end{align}
Furthermore, if the smallest eigenvalue of $\Omega(P)$ is bounded away from zero uniformly over $\mathbf{P}$, $\inf_{P \in \mathbf{P}} \lambda_{\min}(\Omega(P))>0$, then \eqref{eq:conv-ip-to-Jn} holds for $\ell=q$.
\end{lemma}
\begin{proof}
I show the result by way of contradiction. Suppose that the conclusion fails for \(\ell\in\{m,s\}\), and also for \(\ell=q\) under the additional eigenvalue condition. Then, there exist a subsequence $n_l$ and a positive semidefinite correlation matrix $\Omega^*$ such that $\Omega(P_{n_l})\to \Omega^*$ and
\begin{align}\label{contra-hypo:conv-J-to-Jn}
\sup_{x \in \mathbf{R}} \left| J^\ell(x, \sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}), \Omega(\hat{P}_{n_l})) - J^\ell_{n_l}(x)\right| \overset{P_{n_l}}{\not\to} 0.
\end{align}
The existence of $\Omega^*$ and the convergent subsequence follows from $\bar{\mathbf O}$ in being bounded and closed in finite-dimensional Euclidean space and from the Bolzano-Weierstrass theorem.
For $\ell=q$, $\Omega^*$ is positive definite due to the uniformly positive minimum eigenvalue assumption.
This proof handles two cases separately. The first case considers $I=\{1,2,...,k\}$; in the second case, $I$ is a non-empty proper subset of $\{1,2,...,k\}$. Furthermore, we assume \eqref{eq:conv-ip-to-Jn-condition1} as the proof under \eqref{eq:conv-ip-to-Jn-condition2} can be obtained by reversing signs.
\medskip
\noindent
\textbf{Case 1:} Suppose that $I=\{1,2,..., k\}.$
Lemma \ref{alem:consistency-sample-variance} gives $S^{-1}_nD(P_n)=I_k + o_{P_n}(1)$, and hence it implies $\sqrt{n_l}S^{-1}_{n_l} \mu(P_{n_l}) \overset{P_{n_l}}{\to} \delta$. By the uniform CLT in \ref{alem:uniformCLT} and Slutsky's theorem,
$\sqrt{n_l} S^{-1}_{n_l} (\bar{W}_{n_l} - \mu(P_{n_l})) \Rightarrow (\Omega^{*})^{1/2} Z$ for $Z\sim N(0_k, I_k)$ and therefore
\[
\sqrt {n_l} S_{n_l}^{-1}\bar W_{n_l}
=
\sqrt {n_l} S_{n_l}^{-1}(\bar W_{n_l}-\mu(P_{n_l}))
+
\sqrt {n_l} S_{n_l}^{-1}\mu(P_{n_l})
\Rightarrow
\Omega^{*1/2}Z+\delta.
\]
We now handle \(\ell=m\) and \(\ell=s,q\) separately.
\noindent
\emph{The statistic \(T^m\).} Since the map $x\mapsto T^m(x)$ is continuous on \(\mathbf R^k\) and the CDF of $T^m(\Omega^{*1/2}Z+\delta)$ is continuous, the continuous mapping theorem implies that
\begin{align}\label{eq:case-no-div-max1}
J_{n_l}^m(x)
=
P_{n_l}\{
T^m(\sqrt{n_l} S_{n_l}^{-1}\bar W_{n_l})\le x
\}
\to
P\{
T^m(\Omega^{*1/2}Z+\delta)\le x
\} \quad \text{at all }x\in\mathbf{R}.
\end{align}
For an arbitrary deterministic sequence $\theta_{m} \in \Theta$ in $(\mathbf R^k_+\cup \mathbf R^k_-)\times\bar{\mathbf O}$ such that $\theta_{m} \to \theta\in \Theta$ as $m\to\infty$ and for $Z\sim N(0_k, I_k)$, continuity of $T^m$ implies that
\begin{align*}
T^m(\theta_{2,m}^{1/2} Z + \theta_{1,m}) \overset{a.s.}{\to} T^m(\theta_{2}^{1/2} Z + \theta_{1}).
\end{align*}
Since the limit is continuously distributed, it implies the convergence of CDFs at all $x\in\mathbf{R}$:
\begin{align*}
P\{T^m(\theta_{2,m}^{1/2} Z + \theta_{1,m}) \leq x\} \to P\{ T^m(\theta_{2}^{1/2} Z + \theta_{1}) \leq x\}.
\end{align*}
This shows that $J^m(x,\theta_1, \theta_2)=P\{ T^m(\theta_{2}^{1/2} Z + \theta_{1}) \leq x\}$ is continuous in $\theta \in \Theta$ and $x\in \mathbf{R}.$
Recall $\|\Omega(\hat{P}_{n_l}) - \Omega^*\|\overset{P_{n_l}}{\to} 0$ by Lemma \ref{alem:consistency-sample-corr}. The continuity of $J^m(x,\theta_1, \theta_2)$ in $(\theta_1, \theta_2)$ and the continuous mapping theorem combine to give that
\begin{align}\label{eq:case-no-div-max2}
|J^{m}(x, \sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}), \Omega(\hat{P}_{n_l})) - J^m(x, \delta, \Omega^*) |\overset{P_{n_l}}{\to} 0 \quad \text{at all }x\in\mathbf{R}.
\end{align}
Poly\'a's theorem in Theorem 11.2.9 of \cite{LehmannRomano2005testing} applies to both \eqref{eq:case-no-div-max1} and \eqref{eq:case-no-div-max2} as the limit distribution is continuous \footnote{Theorem 11.2.9 states deterministic convergence of CDFs, but the same proof can be extended to show the convergence in probability of CDFs.}. Therefore, combining these yields a contradiction to \eqref{contra-hypo:conv-J-to-Jn}.
\medskip
\noindent
\emph{The statistics \(T^s\) and \(T^q\).}
For $\ell=q$, the following argument is applied on the event $\{\hat\Omega_{n_l}\succ0\}$ so that $T^q_n$, its Gaussian analogue, and the least favorable critical value are all computed using the same matrix $\hat\Omega_{n_l}$. Note that the map $x\mapsto T^s(x)$ is continuous in $\mathbf{R}^k $ and that the map $(y,\Omega)\mapsto T^q(y,\Omega)$ is continuous in $\mathbf{R}^k \times \mathbf{O}.$ For $\ell=s,q$, the same argument above gives
\[
J_{n_l}^\ell(x)
=
P_{n_l}\{
T^\ell(\sqrt{n_l} S_{n_l}^{-1}\bar W_{n_l},\hat{\Omega}_{n_l})\le x
\}
\to
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le x
\}
\]
and
\[
J^\ell(
x,\sqrt{n_l} S_{n_l}^{-1}\mu(P_{n_l}),\hat\Omega_{n_l}
)
\overset{P_{n_l}}{\to}
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le x
\}
\]
for every $x\in(0,\infty)$, since the limiting CDF is continuous on
$(0,\infty)$ by Lemma \ref{lem:strictly_increasing_dist_T}, where the second argument of $T^{s}$ can be ignored, and $\Omega^*\succ0$ for $\ell=q$ due to the uniformly positive eigenvalue assumption.
Furthermore, by Portmanteau Lemma (18.9 (v) in \cite{vandervaart2000asymptotic}),
\begin{align*}
J_{n_l}^\ell(0)=
P_{n_l}\{
\sqrt{n_l} S_{n_l}^{-1}\bar W_{n_l}
&\in
\mathbf R_+^k\cup\mathbf R_-^k
\}\\
&\to
P\{
\Omega^{*1/2}Z+\delta
\in
\mathbf R_+^k\cup\mathbf R_-^k
\} = P\{
T^\ell(\Omega^{*1/2}Z+\delta,\Omega^{*})\le0
\}.
\end{align*}
The same argument applied to the parametric Gaussian analogue gives
\[
J^\ell(
0,\sqrt{n_l} S_{n_l}^{-1}\mu(P_{n_l}),\hat\Omega_{n_l}
)
\overset{P_{n_l}}{\to}
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^{*})\le0
\}.
\]
It remains to upgrade this convergence to uniform convergence. For any $\varepsilon>0$, fix \(\eta>0\) so that $P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le\eta
\}
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le0\} <\varepsilon/4$. Since the limiting CDF is continuous on \((0,\infty)\), by the usual proof of Pólya's theorem, there exists $N_0\in\mathbf{N}$ such that for all $n_l\geq N_0$
\[
\sup_{x\ge\eta}
|
J_{n_l}^\ell(x)
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le x
\}
|
<\tfrac{\varepsilon}{4}.
\]
On \(0\le x\le\eta\), monotonicity of CDFs gives
\[
\begin{aligned}
&\sup_{0\le x\le\eta}
|
J_{n_l}^\ell(x)
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le x
\}
|
\\
&\quad\le
|
J_{n_l}^\ell(0)
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le0
\}
|+
|
J_{n_l}^\ell(\eta)
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le\eta
\}
|\\
&\qquad+
[
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le\eta
\}
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le0
\}]\leq \tfrac34 \varepsilon.
\end{aligned}
\]
Therefore,
\[
\sup_{x\in\mathbf R}
|
J_{n_l}^\ell(x)
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le x
\}
|
\to0.
\]
The same monotonicity argument, replacing \(J_{n_l}^\ell\) by $J^\ell(
x,\sqrt{n_l} S_{n_l}^{-1}\mu(P_{n_l}),\hat\Omega_{n_l})$
gives
\[
\sup_{x\in\mathbf R}
|
J^\ell(
x,\sqrt {n_l} S_{n_l}^{-1}\mu(P_{n_l}),\hat\Omega_{n_l}
)
-
P\{
T^\ell(\Omega^{*1/2}Z+\delta, \Omega^*)\le x
\}
|
\overset{P_{n_l}}{\to}0.
\]
Therefore, combining these two yields a contradiction to \eqref{contra-hypo:conv-J-to-Jn}.
\medskip
\noindent
\textbf{Case 2: \(\varnothing\neq I\subsetneq[k]\).} Assume that $I=\{1,..., \tilde{k}\}$ for some $\tilde{k}<k$ without loss of generality. The proof strategy is analogous to that in \textbf{Case 1}. We continue under \eqref{eq:conv-ip-to-Jn-condition1}. By Lemmas \ref{alem:consistency-sample-variance} and \ref{alem:uniformCLT}, Slutsky's theorem gives
\begin{align}\label{eq:partial-div-to-neg-infty}
\frac{\sqrt{n_l}\bar W_{j,{n_l}}}{S_{j,{n_l}}}
\overset{P_{n_l}}{\to}-\infty \qquad \text{for} \quad j\notin I
\end{align}
and on the coordinates in \(I\)
\begin{align}\label{eq:partial-clt}
(\frac{\sqrt {n_l}\bar W_{j,n_l}}{S_{j,n_l}}
)_{j\in I}
\Rightarrow
(\Omega^*_{I\times I})^{1/2}Z_I+\delta_I,
\qquad
Z_I\sim N(0_{|I|},I_{|I|})
\end{align}
where for a vector \(a\in\mathbf R^k\) we write
\(a_I\) for the subvector indexed by \(I\), and for a matrix \(A\), write
\(A_{I\times I}\) for the principal submatrix indexed by \(I\).
We now handle \(\ell=m\) and \(\ell=s,q\) separately.
\noindent
\emph{The statistic \(T^m\).} First, I derive the convergence of
\(J(x,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l}))\).
Consider an arbitrary deterministic sequence
\(\theta_m=(\theta_{1,m},\theta_{2,m})\in\Theta\) in $[-\infty,0]^k\times\bar{\mathbf O}$ such that $\theta_{1,j,m}\to \delta_j$ for $j\in I$, $\theta_{1,j,m}\to-\infty$ for $j\notin I$, and $\theta_{2,m}\to\Omega^*.$ Thus, for every \(Z\sim N(0_k, I_k)\),
\[
T^m(\theta_{2,m}^{1/2}Z+\theta_{1,m})
\overset{a.s.}{\to}
\max_{j\in I}
\{((\Omega^*)^{1/2}Z)_j+\delta_j\}
\]
because $(\theta_{2,m}^{1/2}Z+\theta_{1,m})_I \overset{a.s.}{\to}
((\Omega^*)^{1/2}Z)_I+\delta_I,$ and $(\theta_{2,m}^{1/2}Z+\theta_{1,m})_j \overset{a.s.}{\to} -\infty$ for all $j\notin I.$ Since the limit distribution is continuous, this implies the convergence of the associated CDFs at all $x\in\mathbf{R}$:
\[
P\{T^m(\theta_{2,m}^{\tfrac12}Z+\theta_{1,m})\leq x\}
\to
P\{\max_{j\in I}
\{((\Omega^*)^{\tfrac12}Z)_j+\delta_j\} \leq x \}
= P\{\max_{j\in I}
\{((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j\} \leq x \}
\]
where the equality holds because $((\Omega^*)^{1/2}Z)_I
\overset{d}{=}
(\Omega^*_{I\times I})^{1/2}Z_I$. By Pólya's theorem,
\[
\begin{aligned}
&\textstyle\sup_{x\in\mathbf R}
|
P\{
T^m(\theta_{2,m}^{1/2}Z+\theta_{1,m})\le x
\}
-
P\{
\textstyle\max_{j\in I}
[
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j
]
\le x
\}
|
\to0.
\end{aligned}
\]
Now apply this deterministic conclusion to the random sequence $\theta_{1,n_l}:=\sqrt{n_l}S_{n_l}^{-1}\mu(P_{n_l})$, $\theta_{2,n_l}:=\Omega(\hat P_{n_l}).$
By Lemmas \ref{alem:consistency-sample-variance} and \ref{alem:consistency-sample-corr}, $\theta_{1,n_l,I}\overset{P_{n_l}}{\to}\delta_I$, $\max_{j\notin I}\theta_{1,j,n_l}
\overset{P_{n_l}}{\to}-\infty$, and $\theta_{2,n_l}\overset{P_{n_l}}{\to}\Omega^*.$
Therefore, by the deterministic convergence just established and the subsequence principle,
\begin{align}
\label{eq:conv-ip-to-Jn-partialI-part1}
&\sup_{x\in\mathbf R}
|
J^m(
x,\sqrt{n_l}S_{n_l}^{-1}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)
-
P\{
\max_{j\in I}
[
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j
]
\le x
\}
|
\overset{P_{n_l}}{\to}0.
\end{align}
Second, I consider the convergence of \(J_{n_l}^m(x)\). Define
\[
X_{n_l}=(X_{n_l,1},..., X_{n_l,k}):=\sqrt{n_l}S_{n_l}^{-1}\bar W_{n_l}.
\]
Then the probability of the following event converges to 1,
\begin{align*}
P\{G_{n_l}\} :=P\{\max_{j\in I}X_{n_l,j}
\ge
\max_{j\notin I}X_{n_l,j},
\min_{j\notin I}X_{n_l,j}
\le
\min_{j\in I}X_{n_l,j},
\max_{j\in I}X_{n_l,j}
\le
\max_{j\notin I}\{-X_{n_l,j}\}\} \to 1
\end{align*}
because of \eqref{eq:partial-div-to-neg-infty}, \eqref{eq:partial-clt} and Slutsky's theorem. On \(G_{n_l}\), $T^m(X_{n_l})
=
\max_{j\in I} X_{n_l,j}$ and therefore $T^m(X_{n_l})
-
\max_{j\in I}(X_{n_l})_j
\overset{P_{n_l}}{\to}0$, which implies weak convergence. Since the limiting distribution is continuous, Pólya's theorem yields
\begin{align}
\label{eq:conv-ip-to-Jn-partialI-part2}
&\sup_{x\in\mathbf R}
|
J_{n_l}^m(x)
-
P\{
\max_{j\in I}
[
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j
]
\le x
\}
|
\to0.
\end{align}
Finally, combining \eqref{eq:conv-ip-to-Jn-partialI-part1} and
\eqref{eq:conv-ip-to-Jn-partialI-part2}, we obtain
\[
\begin{aligned}
&\sup_{x\in\mathbf R}
|
J^m(
x,\sqrt{n_l}S_{n_l}^{-1}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)
-
J_{n_l}^m(x)
|
\overset{P_{n_l}}{\to}0,
\end{aligned}
\]
which contradicts \eqref{contra-hypo:conv-J-to-Jn}.
\medskip
\noindent
\emph{The statistic \(T^s\).} First, I derive the convergence of
\(J^s(x,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l}))\).
Consider the deterministic sequence
\(\theta_m=(\theta_{1,m},\theta_{2,m})\in\Theta\). Then, for every \(Z\sim N(0_k,I_k)\), $(\theta_{2,m}^{1/2}Z+\theta_{1,m})_I
\overset{a.s.}{\to}
((\Omega^*)^{1/2}Z)_I+\delta_I$ and $(\theta_{2,m}^{1/2}Z+\theta_{1,m})_j
\overset{a.s.}{\to}
-\infty$ for all $j\notin I.$
Hence
\[
T^s(\theta_{2,m}^{1/2}Z+\theta_{1,m})
\overset{a.s.}{\to}
\sum_{j\in I}
(((\Omega^*)^{1/2}Z)_j+\delta_j)^2
I\{((\Omega^*)^{1/2}Z)_j+\delta_j\ge0\}.
\]
Since \(((\Omega^*)^{1/2}Z)_I\overset{d}{=}(\Omega^*_{I\times I})^{1/2}Z_I\),
it follows that for any $x\in(0,\infty)$
\[
P\{ T^s(\theta_{2,m}^{\tfrac12}Z+\theta_{1,m}) \leq x\}
\to P\left\{
\sum_{j\in I}
[
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j\ge0
\} \leq x\right\}
\]
as limiting CDF is continuous on \((0,\infty)\). By the same argument as in the proof for \(T^m\), applied to the random sequence
\(\theta_{1,n_l}:=\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l})\) and
\(\theta_{2,n_l}:=\Omega(\hat P_{n_l})\), we have, at every continuity point \(x>0\) of the limiting CDF,
\[
\begin{aligned}
&J^s(
x,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)
\overset{P_{n_l}}{\to}
P\bigg\{
\sum_{j\in I}
[
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j\ge0
\}
\le x
\bigg\}.
\end{aligned}
\]
We now verify convergence at zero. Since \(T^s(y)=0\) if and only if
\(y\in\mathbf R_+^k\cup\mathbf R_-^k\), and since the coordinates outside \(I\)
diverge to \(-\infty\), the positive-orthant event becomes asymptotically impossible.
Thus
\[
\begin{aligned}
J^s(
0,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)&\overset{P_{n_l}}{\to}
P\{
(\Omega^*_{I\times I})^{\tfrac12}Z_I+\delta_I\in\mathbf R_-^{|I|}
\}\\
&=P\bigg\{
\sum_{j\in I}
[
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j\ge0
\}
\le0
\bigg\}.
\end{aligned}
\]
Next, I consider the convergence of \(J^s_{n_l}(x)\). Recall
\(X_{n_l}:=\sqrt{n_l}S^{-1}_{n_l}\bar W_{n_l}\). We can show
\[
T^s(X_{n_l})
\Rightarrow
\textstyle\sum_{j\in I}
[
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j\ge0
\}
\]
because $T^s(X_{n_l}) -\sum_{j\in I} X^2_{n_l, j} I\{X_{n_l,j}\geq 0\}\overset{P_{n_l}}{\to} 0$ and by the continuous mapping theorem.
At zero, we have
\begin{align*}
J^s_{n_l}(0)
&\overset{(a)}{=}
P_{n_l}\{
X_{n_l}\in\mathbf R_+^k\cup\mathbf R_-^k\}\overset{(b)}{=}
P_{n_l}\{(X_{n_l})_I\in\mathbf R_-^{|I|}\} + o(1)
\overset{(c)}{\to}
P\{
(\Omega^*_{I\times I})^{1/2}Z_I+\delta_I\in\mathbf R_-^{|I|}
\}
\end{align*}
where (a) holds because \(T^s(y)=0\) if and only if
\(y\in\mathbf R_+^k\cup\mathbf R_-^k\), (b) holds because \eqref{eq:partial-div-to-neg-infty}, and (c) follows from the Portmanteau lemma.
It remains to upgrade the convergence to uniform convergence. Repeating the argument used in \textbf{Case 1}, we can show
\[
\begin{aligned}
&\sup_{x\in\mathbf R}
|
J^s_{n_l}(x)
-
P\{
\textstyle\sum_{j\in I}
[
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j\ge0
\}
\le x
\}
|
\to0,\\
&\sup_{x\in\mathbf R}
\left|
\begin{aligned}
&J^s(
x,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l}))
\\
&\qquad-P\{
\textstyle\sum_{j\in I}
[
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{\tfrac12}Z_I)_j+\delta_j\ge0
\}
\le x
\}
\end{aligned}
\right|
\overset{P_{n_l}}{\to}0.
\end{aligned}
\]
Combining these two displays gives a contradiction to \eqref{contra-hypo:conv-J-to-Jn}.
\medskip
\noindent
\emph{The statistic \(T^q\).} Now impose the additional assumption that $\inf_{P\in\mathbf P}\lambda_{\min}(\Omega(P))>0.$ Then $\Omega^*\succ0$. Since $P_{n_l}\{\hat{\Omega}_{n_l}\not\succ0\}\to0$ by Lemma \ref{lem:sample-corr-positive-definite}, it suffices to establish the following convergences on the event $\{\hat{\Omega}_{n_l}\succ0\}$. We use the same argument used for \(T^s\). The only difference is that, when the coordinates outside \(I\) diverge to \(-\infty\), the statistic \(T^q\) reduces to the quadratic distance from the limiting \(I\)-subvector to \(\mathbf R_-^{|I|}\). More precisely, the deterministic reduction corresponding to the one used for \(T^s\) gives
\[
T^q(\theta_{2,m}^{1/2}Z+\theta_{1,m},\theta_{2,m})
\Rightarrow
\inf_{\nu\in\mathbf R_-^{|I|}}
((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu)'
(\Omega^*_{I\times I})^{-1}
((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu)
\]
whenever \(\theta_{1,j,m}\to\delta_j\) for \(j\in I\), \(\theta_{1,j,m}\to-\infty\) for \(j\notin I\), and \(\theta_{2,m}\to\Omega^*\). Applying this deterministic reduction to $\theta_{1,n_l}:=\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),$ and $\theta_{2,n_l}:=\Omega(\hat P_{n_l}),$ gives, at every continuity point \(x>0\) of the limiting CDF,
\[
\begin{aligned}
&J^q(
x,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)
\\
&\quad\overset{P_{n_l}}{\to}
P\bigg\{
\inf_{\nu\in\mathbf R_-^{|I|}}
\bigl((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu\bigr)'
(\Omega^*_{I\times I})^{-1}
\bigl((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu\bigr)
\le x
\bigg\}.
\end{aligned}
\]
The same reduction applied to \(X_{n_l}=\sqrt{n_l}S^{-1}_{n_l}\bar W_{n_l}\) gives
\[
\begin{aligned}
T^q(X_{n_l},\Omega(\hat P_{n_l}))
\Rightarrow
\textstyle\inf_{\nu\in\mathbf R_-^{|I|}}
((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu\bigr)'
(\Omega^*_{I\times I})^{-1}
((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu\bigr).
\end{aligned}
\]
At zero, the reduced statistic equals zero if and only if $(\Omega^*_{I\times I})^{1/2}Z_I+\delta_I\in\mathbf R_-^{|I|}.$ Thus, by the same cone-event argument used for \(T^s\),
\[
\begin{aligned}&
J^q_{n_l}(0)
\to
P\{
(\Omega^*_{I\times I})^{1/2}Z_I+\delta_I\in\mathbf R_-^{|I|}
\},\\
&J^q(
0,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)
\overset{P_{n_l}}{\to}
P\{
(\Omega^*_{I\times I})^{1/2}Z_I+\delta_I\in\mathbf R_-^{|I|}
\}.\end{aligned}
\]
Since the limiting CDF is continuous on \((0,\infty)\), the same split-at-zero argument used for \(T^s\) yields
\[
\begin{aligned}
&\sup_{x\in\mathbf R}
|
J^q(
x,\sqrt{n_l}S^{-1}_{n_l}\mu(P_{n_l}),\Omega(\hat P_{n_l})
)
-
J^q_{n_l}(x)
|
\overset{P_{n_l}}{\to}0.
\end{aligned}
\]
This contradicts \eqref{contra-hypo:conv-J-to-Jn}.
Finally, suppose \eqref{eq:conv-ip-to-Jn-condition2} holds. The proof is identical
after replacing \(W_i\) by \(-W_i\). In the partial-divergence case, the coordinates
outside \(I\) diverge to \(+\infty\). Accordingly, the reduced limiting statistic for
\(T^m\) becomes
\[
\max_{j\in I}
\{-((\Omega^*_{I\times I})^{1/2}Z_I)_j-\delta_j\},
\]
the reduced limiting statistic for \(T^s\) becomes
\[
\sum_{j\in I}
[
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j
]^2
I\{
((\Omega^*_{I\times I})^{1/2}Z_I)_j+\delta_j\le0
\},
\]
and, under the eigenvalue condition, the reduced limiting statistic for \(T^q\) becomes
\[
\inf_{\nu\in\mathbf R_+^{|I|}}
\bigl((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu\bigr)'
(\Omega^*_{I\times I})^{-1}
\bigl((\Omega^*_{I\times I})^{1/2}Z_I+\delta_I-\nu\bigr).
\]
At zero, the relevant cone event is $(\Omega^*_{I\times I})^{1/2}Z_I+\delta_I\in\mathbf R_+^{|I|}.$ The same continuity, Portmanteau, Pólya, and split-at-zero arguments then give the desired contradiction.
\end{proof}
\begin{lemma}\label{lem:sample-corr-positive-definite}
Consider any sequence $\{P_n\in\mathbf P:n\ge1\}$, where $\mathbf P$ satisfies \eqref{def:uniform-integrability}. If
$$
\lambda^*
:=
\inf_{P\in\mathbf P}\lambda_{\min}(\Omega(P))
>0,
$$
then
$$
P_n\{\hat\Omega_n\succ0\}\to1.
$$
\end{lemma}
\begin{proof}
By Lemma \ref{alem:consistency-sample-corr},
$
\|\hat\Omega_n-\Omega(P_n)\|
\overset{P_n}{\to}0,
$
where $\|\cdot\|$ is the componentwise maximum norm. Since
$\|A\|_{\mathrm{op}}\le k\|A\|$, Weyl's inequality gives $\lambda_{\min}(\hat\Omega_n)
\ge
\lambda_{\min}(\Omega(P_n))
-
\|\hat\Omega_n-\Omega(P_n)\|_{\mathrm{op}} \ge
\lambda^*
-
k\|\hat\Omega_n-\Omega(P_n)\|.$ Since $\{\lambda_{\min}(\hat\Omega_n)>\lambda^*/2\} \subseteq\{\hat\Omega_n\succ0\},$ it holds that
$$
P_n\{\hat\Omega_n\not\succ0\}
\le
P_n\{\lambda_{\min}(\hat\Omega_n)\leq \lambda^*/2\}
\le
P_n\{
\|\hat\Omega_n-\Omega(P_n)\|
\ge{\lambda^*}/{2k}
\}
\to0.
$$
\end{proof}
\begin{lemma}\label{lem:strictly_increasing_dist_T}
Let $k\ge2$ and $\theta_1\in\mathbf R^k$.
\begin{enumerate}[i)]
\item For $\ell=m$ and $\theta_2\in\bar{\mathbf O}$,
$J^m(\cdot,\theta_1,\theta_2)$ is continuous and strictly increasing
wherever it takes values in $(0,1)$.
\item For $\ell=s$ and $\theta_2\in\bar{\mathbf O}$, and for $\ell=q$ and
$\theta_2\in\mathbf O$, the distribution of $T^\ell(\theta_2^{1/2}Z+\theta_1,\theta_2)$ may have an atom at zero, but its CDF is continuous on $(0,\infty)$ and
strictly increasing at every $x>0$ where the second argument of $T^s$ is ignored.
\end{enumerate}
\end{lemma}
\begin{proof}
Let \(r=\operatorname{rank}(\theta_2)\). Choose \(A\in\mathbf R^{k\times r}\)
with rank \(r\) such that \(AA'=\theta_2\). Then
\(\theta_2^{1/2}Z\stackrel{d}{=}A\xi\), where
\(\xi\sim N(0,I_r)\). Since \(\theta_2\) is a correlation matrix, each row
\(a_j'\) of \(A\) satisfies \(\|a_j\|=1\). Define $h(\xi):=T^m(A\xi+\theta_1)$. Then $J^m(x,\theta_1,\theta_2)=P\{h(\xi)\le x\}.$
We first show that \(P\{h(\xi)=t\}=0\) for every \(t\in\mathbf R\). The
function \(h\) is continuous and piecewise affine. Indeed, $h(\xi)
=
\min\left\{
\max_{1\le j\le k}(a_j'\xi+\theta_{1,j}),
\max_{1\le j\le k}(-a_j'\xi-\theta_{1,j})
\right\}.$ Hence \(\mathbf R^r\) can be partitioned into finitely many polyhedra such that, on each polyhedron, \(h\) is equal to one of the affine functions \(a_j'\xi+\theta_{1,j}\) or \(-a_j'\xi-\theta_{1,j}\). Since
\(\|a_j\|=1\), none of these affine functions is constant. Therefore, on each
polyhedron, the level set \(\{\xi:h(\xi)=t\}\) is contained in an affine
hyperplane. Since there are only finitely many such polyhedra,
\(\{\xi:h(\xi)=t\}\) has Lebesgue measure zero. Because \(\xi\) has a density
on \(\mathbf R^r\), it follows that \(P\{h(\xi)=t\}=0\) for every \(t\in\mathbf R\) so \(J^m(\cdot,\theta_1,\theta_2)\) is continuous.
It remains to prove strict monotonicity on the non-degenerate part of the
support. Let $x$ satisfy $0<J(x,\theta_1,\theta_2)<1$. Since $h$ is continuous and $\mathbf{R}^r$ is connected, $h(\mathbf{R}^r)$ is an interval in $\mathbf{R}$. Also, $J(x,\theta_1,\theta_2)<1$ implies that there exists $y\in h(\mathbf{R}^r)$ with $y>x$. Hence, for every $\varepsilon\in(0,y-x)$, $h(\mathbf{R}^r)\cap(x,x+\varepsilon)\neq\varnothing.$ Therefore $h^{-1}((x,x+\varepsilon))$ is a nonempty open subset of $\mathbf{R}^r$. Since $\xi$ has a strictly positive density on $\mathbf{R}^r$, $P\{x<h(\xi)\le x+\varepsilon\}>0,$ and so $J^m(x+\varepsilon,\theta_1,\theta_2)>J^m(x,\theta_1,\theta_2).$ Thus $J^m(\cdot,\theta_1,\theta_2)$ is strictly increasing at every $x$ such that $0<J^m(x,\theta_1,\theta_2)<1$.
For $\ell\in\{s,q\}$, the proof is analogous after accounting for the
possible atom at zero. Using the representation
$\theta_2^{1/2}Z\overset{d}{=}A\xi$, the corresponding statistic is a
continuous piecewise-quadratic function of $\xi$. On each piece on which
the statistic is positive, it is a nonconstant quadratic function.
Consequently, every strictly positive level set has Lebesgue measure
zero, so the distribution has no atoms on $(0,\infty)$. Moreover, the
image of the connected set $\mathbf R^r$ under this continuous function
is an interval. Hence, whenever
$J^\ell(0,\theta_1,\theta_2)
<
J^\ell(x,\theta_1,\theta_2)
<
1,$ the inverse image of $(x,x+\varepsilon)$ is a nonempty open set for every
sufficiently small $\varepsilon>0$. Since $\xi$ has a strictly positive
density, this set has positive probability, proving strict increase.
\end{proof}
\subsubsection{Lemmas for Corollary \ref{cor:LFCtest-nonnegative-cv}}
\begin{lemma}\label{lem:int-area}
For any \(x\in\mathbf R\) and any non-empty set \(\Lambda\subseteq[k]\), define
\[
\mathbf A(x,\Lambda)
:=
\left\{
w_\Lambda\in\mathbf R^{|\Lambda|}:
\exists i,j\in\Lambda \text{ such that } w_i>x \text{ and } w_j<-x
\right\}.
\]
For simplicity, write
\[
\mathbf A_k(x):=\mathbf A(x,[k]).
\]
Consider \(T^m\) defined in \eqref{def:T-ell-function}. If \(k\ge2\), then the following statements hold.
\begin{enumerate}[(i)]
\item \(\mathbf A_k(x)\) is strictly decreasing in \(x\): for any \(x'>x\), $ \mathbf A_k(x')\subsetneq \mathbf A_k(x).$
\item $ \{w\in\mathbf R^k:T^m(w)>x\}=\mathbf A_k(x).$
\item \(\mathbf A_k(x)\) is symmetric: if \(w\in\mathbf A_k(x)\), then
\(-w\in\mathbf A_k(x)\).
\end{enumerate}
\end{lemma}
\begin{proof}
\textit{(i)} If \(w\in\mathbf A_k(x')\), then there exist \(i,j\in[k]\) such that
\(w_i>x'\) and \(w_j<-x'\). Since \(x'>x\), this implies \(w_i>x\) and
\(w_j<-x\). Hence \(\mathbf A_k(x')\subseteq \mathbf A_k(x)\). The inclusion is strict. For example, take $w_1=\frac{x+x'}{2},$ $w_2=-\frac{x+x'}{2}$, and set the remaining coordinates arbitrarily. Then \(w_1>x\) and \(w_2<-x\),
so \(w\in\mathbf A_k(x)\). However, \(w_1\le x'\) and \(w_2\ge -x'\), so
\(w\notin\mathbf A_k(x')\). Therefore
\(\mathbf A_k(x')\subsetneq \mathbf A_k(x)\).
\textit{(ii)} By definition of \(T^m\),
\begin{align*}
\{w\in\mathbf R^k:T^m(w)>x\}
&=
\{
w\in\mathbf R^k:
\min\{
\max_{1\le j\le k}w_j,
\max_{1\le j\le k}(-w_j)
\}>x
\} \\
&=
\{
w\in\mathbf R^k:
\max_{1\le j\le k}w_j>x
\text{ and }
\max_{1\le j\le k}(-w_j)>x
\} \\
&=
\{
w\in\mathbf R^k:
\exists i,j\in[k]\text{ such that }w_i>x
\text{ and }w_j<-x
\} \\
&=\mathbf A_k(x).
\end{align*}
\textit{(iii)} Let \(w\in\mathbf A_k(x)\). Then there exist \(i,j\in[k]\) such
that \(w_i>x\) and \(w_j<-x\). Hence \(-w_i<-x\) and \(-w_j>x\), so
\(-w\in\mathbf A_k(x)\).
\end{proof}
\begin{lemma}\label{lem:monotonedecreasing-in-theta2j}
Let $\theta_1\in \mathbf R_+^k$ and $\theta_2\in \mathbf O$. Consider $J^m$ in \eqref{def:J}. Then for any fixed $x\in \mathbf R$ and any
$j\in \{(a,b)\in [k]\times [k]:a<b\}$,
\[
\frac{\partial}{\partial \theta_{2,j}} J^m(x,\theta_1,\theta_2)>0.
\]
\end{lemma}
\begin{proof}
Without loss of generality, it suffices to prove the claim for
$j=(1,2)$. We consider two cases where $x<0$ and $x\geq 0.$
\noindent
\textit{Case 1: $x<0$.}
For simplicity, let $Y:=\theta_2^{1/2}Z+\theta_1$ where $Z \sim N(0_k, I_k)$. Since
\begin{align*}
\{T^m(y)\le x\}
&=
\{\max_{1\le m\le k} y_m\le x\}
\cup
\{\max_{1\le m\le k} (-y_m)\le x\}\\
&=
\{y_m\le x \text{ for all }m\}
\cup
\{y_m\ge -x \text{ for all }m\}
\end{align*}
and the two unioning sets are exclusive, we have
\begin{align*}
J^m(x,\theta_1,\theta_2)
&=
P\{Y_m\le x \text{ for all }m\}
+
P\{Y_m\ge -x \text{ for all }m\}.
\end{align*}
Further, write $u_m:=x-\theta_{1,m},$ and $v_m:=-x-\theta_{1,m}$ for $m=1,\dots,k.$
Then,
\begin{align*}
\frac{\partial}{\partial \theta_{2,(1,2)}}
P\{Y_m\le x \text{ for all }m\}
&\overset{(a)}{=}
\frac{\partial}{\partial \theta_{2,(1,2)}}\int_{-\infty}^{u_k}... \int_{-\infty}^{u_3}
\int_{-\infty}^{u_2}\int_{-\infty}^{u_1} \phi_k(t;\theta_2)\,
dt_1dt_2dt_3... dt_k\\
&\overset{(b)}{=}
\int_{-\infty}^{u_k}... \int_{-\infty}^{u_3}
\int_{-\infty}^{u_2}\int_{-\infty}^{u_1} \frac{\partial}{\partial \theta_{2,(1,2)}}\phi_k(t;\theta_2)\,
dt_1dt_2dt_3... dt_k\\
&\overset{(c)}{=}
\int_{-\infty}^{u_k}... \int_{-\infty}^{u_3}
\int_{-\infty}^{u_2}\int_{-\infty}^{u_1}
\frac{\partial^2}{\partial t_2\partial t_1}\phi_k(t;\theta_2)\,
dt_1dt_2dt_3... dt_k\\
&\overset{(d)}{=}
\int_{-\infty}^{u_k}... \int_{-\infty}^{u_3}
\phi_k(u_1,u_2,t_3,\dots,t_k;\theta_2)\,dt_3... dt_k>0
\end{align*}
where (a) holds by definition; (b) holds by interchanging differentiation and integration, which is justified by the dominated convergence theorem, since $\theta_2\in \mathbf O$ is positive definite, the map $(t,\Sigma)\mapsto \phi_k(t;\Sigma)$ is continuously differentiable in $\Sigma$ on a neighborhood of $\theta_2$, and the derivative $\partial \phi_k(t;\Sigma)/\partial \Sigma_{(1,2)}$ is dominated on that neighborhood by an integrable function of $t$; (c) holds by applying Plackett's identity $\frac{\partial}{\partial \theta_{2,(1,2)}}\phi_k(t;\theta_2) = \frac{\partial^2}{\partial t_2\partial t_1}\phi_k(t;\theta_2)$; and (d) follows from the fundamental theorem of calculus.
A similar argument gives $\frac{\partial}{\partial \theta_{2,(1,2)}}
P\{Y_m\ge -x \text{ for all }m\}>0$. Therefore
\[
\frac{\partial}{\partial \theta_{2,(1,2)}}J^m(x,\theta_1,\theta_2)>0
\qquad\text{for all }x<0.
\]
\noindent
\textit{Case 2: $x\ge 0$.} Let $b_m^-:=-x+\theta_{1,m}$ and $b_m^+:=x+\theta_{1,m},$ for $m=1,\dots,k.$ Since $x\ge 0$, we have $b_m^-\le b_m^+$ for every $m$.
By Lemma \ref{lem:int-area},
\[
\{y: T^m(y)>x\}=\mathbf A_k(x)-\theta_1.
\]
Because \(\mathbf A_k(x)\) is symmetric,
\(\mathbf A_k(x)=-\mathbf A_k(x)\), and because the centered Gaussian
density is even, \(\phi_k(-t;\theta_2)=\phi_k(t;\theta_2)\), the change
of variables \(s=-t\) gives
\[
\begin{aligned}
1-J^m(x,\theta_1,\theta_2)
&=
\int_{\mathbf A_k(x)-\theta_1}
\phi_k(t;\theta_2)\,dt
=
\int_{\mathbf A_k(x)+\theta_1}
\phi_k(s;\theta_2)\,ds.
\end{aligned}
\]
Moreover,
\[
\mathbf A_k(x)+\theta_1
=
\bigcup_{i\ne j}
\{t\in\mathbf R^k:t_i<b_i^-,\ t_j>b_j^+\}.
\]
As before, by interchanging differentiation and integration and applying Plackett's identity, we have
\begin{align}
\frac{\partial}{\partial \theta_{2,(1,2)}}(1-J^m(x,\theta_1,\theta_2))
=
\int_{t\in \mathbf A_k(x)+\theta_1}
\frac{\partial}{\partial \theta_{2,(1,2)}}\phi_k(t;\theta_2)\,dt
=
\int_{t\in \mathbf A_k(x)+\theta_1}
\frac{\partial^2}{\partial t_2\partial t_1}\phi_k(t;\theta_2)\,dt.
\label{eq:main-derivative-proof}
\end{align}
\noindent
For fixed $(t_2,\dots,t_k)\in \mathbf R^{k-1}$, the fundamental theorem of calculus implies
\begin{align}
\int_{\mathbf R}\frac{\partial^2}{\partial t_2\partial t_1}\phi_k(t;\theta_2)\,dt_1
&=
\lim_{t_1\to\infty}\frac{\partial}{\partial t_2}\phi_k(t;\theta_2)
-
\lim_{t_1\to-\infty}\frac{\partial}{\partial t_2}\phi_k(t;\theta_2)
=0,
\label{eq:prelim1}
\end{align}
because
\[
\frac{\partial}{\partial t_i}\phi_k(t;\theta_2)
=
-\phi_k(t;\theta_2)\sum_{m=1}^k \theta^{-1}_{2,(i,m)}t_m,
\]
and the Gaussian density and its first derivatives vanish at infinity. Likewise, for any $a<b$,
\begin{align}
\int_{\mathbf R}\int_a^b
\frac{\partial^2}{\partial t_2\partial t_1}\phi_k(t;\theta_2)\,dt_1dt_2
&=
\int_{\mathbf R}
\left[
\frac{\partial}{\partial t_2}\phi_k((b,t_2,...,t_k);\theta_2)
-
\frac{\partial}{\partial t_2}\phi_k((a,t_2,...,t_k);\theta_2)
\right]dt_2 =0.
\label{eq:prelim2}
\end{align}
\noindent
\textit{Case 2(a): $k=2$.} If $k=2$, $\mathbf A_2(x)+\theta_1
=
(-\infty,b_1^-)\times (b_2^+,\infty)
\;\cup\;
(b_1^+,\infty)\times (-\infty,b_2^-).$ Therefore,
\begin{align*}
\frac{\partial}{\partial \theta_{2,(1,2)}}\bigl(1-J^m(x,\theta_1,\theta_2)\bigr)
&=
\int_{b_2^+}^{\infty}\int_{-\infty}^{b_1^-}
\frac{\partial^2}{\partial t_2\partial t_1}\phi_2(t;\theta_2)\,dt_1dt_2+
\int_{-\infty}^{b_2^-}\int_{b_1^+}^{\infty}
\frac{\partial^2}{\partial t_2\partial t_1}\phi_2(t;\theta_2)\,dt_1dt_2\\
&=
-\phi_2(b_1^-,b_2^+;\theta_2)-\phi_2(b_1^+,b_2^-;\theta_2)<0.
\end{align*}
Hence
\[
\frac{\partial}{\partial \theta_{2,(1,2)}}J^m(x,\theta_1,\theta_2)>0
\qquad\text{for }x\ge 0,\ k=2.
\]
\noindent
\textit{Case 2(b): $k\ge 3$.} Fix $t_{3:k}:=(t_3,\dots,t_k)\in \mathbf R^{k-2}$ and define the section
\[
B(t_{3:k})
:=
\{(t_1,t_2)\in \mathbf R^2:(t_1,t_2,t_{3:k})\in \mathbf A_k(x)+\theta_1\}.
\]
For each fixed $t_{3:k}$, exactly one of the following four cases occurs.
\begin{enumerate}[i)]
\item There exist $m,\ell\in\{3,...,k\}$ such that
$t_m<b_m^-$ and $t_\ell>b_\ell^+$. Then $B(t_{3:k})=\mathbf R^2. $
\item There exists $m\in\{3,...,k\}$ such that
$t_m<b_m^-$ and $t_\ell\le b_\ell^+$ for all $\ell=3,...,k$. Then, $B(t_{3:k})=\{t_1>b_1^+\}\cup\{t_2>b_2^+\}.$
\item There exists $m\in\{3,...,k\}$ such that
$t_m>b_m^+$ and $t_\ell\ge b_\ell^-$ for all $\ell=3,...,k$. Then, $B(t_{3:k})=\{t_1<b_1^-\}\cup\{t_2<b_2^-\}.$
\item For all $\ell=3,...,k$, $b_\ell^-\le t_\ell\le b_\ell^+.$ Then, $B(t_{3:k})
=
\{t_1<b_1^-,\,t_2>b_2^+\}
\cup
\{t_1>b_1^+,\,t_2<b_2^-\}.$
\end{enumerate}
For simplicity of notation, let $g_{t_{3:k}}(t_1,t_2)
:=
\phi_k(t_1,t_2,t_3,\dots,t_k;\theta_2).$ We compute, case by case, the section integral
\[
I(t_{3:k})
:=
\iint_{B(t_{3:k})}
\frac{\partial^2}{\partial t_2\partial t_1}
g_{t_{3:k}}(t_1,t_2)\,dt_1dt_2.
\]
\noindent
In case i), by \eqref{eq:prelim1}--\eqref{eq:prelim2},
\[
I(t_{3:k})
=
\int_{\mathbf R}\int_{\mathbf R}
\frac{\partial^2}{\partial t_2\partial t_1}
g_{t_{3:k}}(t_1,t_2)\,dt_1dt_2
=0.
\]
\noindent
In case ii), $B(t_{3:k})=\mathbf R^2\setminus \bigl((-\infty,b_1^+]\times (-\infty,b_2^+]\bigr),$ so
\begin{align*}
I(t_{3:k})
&=
-\int_{-\infty}^{b_2^+}\int_{-\infty}^{b_1^+}
\frac{\partial^2}{\partial t_2\partial t_1}
g_{t_{3:k}}(t_1,t_2)\,dt_1dt_2
=
-g_{t_{3:k}}(b_1^+,b_2^+)
=
-\phi_k(b_1^+,b_2^+,t_{3:k};\theta_2)<0.
\end{align*}
\noindent
In case iii), $B(t_{3:k})=\mathbf R^2\setminus \bigl([b_1^-,\infty)\times [b_2^-,\infty)\bigr),$ so
\begin{align*}
I(t_{3:k})
&=
-\int_{b_2^-}^{\infty}\int_{b_1^-}^{\infty}
\frac{\partial^2}{\partial t_2\partial t_1}
g_{t_{3:k}}(t_1,t_2)\,dt_1dt_2
=
-g_{t_{3:k}}(b_1^-,b_2^-)
=
-\phi_k(b_1^-,b_2^-,t_{3:k};\theta_2)<0.
\end{align*}
\noindent
In case iv),
\begin{align*}
I(t_{3:k})
&=
\int_{b_2^+}^{\infty}\int_{-\infty}^{b_1^-}
\frac{\partial^2}{\partial t_2\partial t_1}
g_{t_{3:k}}(t_1,t_2)\,dt_1dt+
\int_{-\infty}^{b_2^-}\int_{b_1^+}^{\infty}
\frac{\partial^2}{\partial t_2\partial t_1}
g_{t_{3:k}}(t_1,t_2)\,dt_1dt_2\\
&=
-g_{t_{3:k}}(b_1^-,b_2^+)-g_{t_{3:k}}(b_1^+,b_2^-)=
-\phi_k(b_1^-,b_2^+,t_{3:k};\theta_2)
-\phi_k(b_1^+,b_2^-,t_{3:k};\theta_2)
<0.
\end{align*}
Thus, for every fixed $t_{3:k}$, $I(t_{3:k})\le 0, $ and moreover $I(t_{3:k})<0$ unless case (i) occurs. Integrating the section integrals over $t_{3:k}$ and using
\eqref{eq:main-derivative-proof}, I obtain
\[
\frac{\partial}{\partial \theta_{2,(1,2)}}\bigl(1-J^m(x,\theta_1,\theta_2)\bigr)
=
\int_{\mathbf R^{k-2}} I(t_{3:k})\,dt_{3:k}\le 0.
\]
It remains to prove strict inequality. If $x>0$, then the set $\prod_{\ell=3}^k [b_\ell^-,b_\ell^+]$ has positive Lebesgue measure, and for every $t_{3:k}$ in this set, case iv) occurs. Hence $\int_{\mathbf R^{k-2}} I(t_{3:k})\,dt_{3:k}<0.$ If $x=0$, then $b_\ell^-=b_\ell^+=\theta_{1,\ell}$, so case iv) occurs only on
a null set. However, the set
\[
\{t_{3:k}: t_\ell\le \theta_{1,\ell}\ \forall \ell=3,\dots,k,\ \text{and }t_m<\theta_{1,m}\text{ for some }m\}
\]
has positive Lebesgue measure, and on this set case ii) occurs; likewise, the
set
\[
\{t_{3:k}: t_\ell\ge \theta_{1,\ell}\ \forall \ell=3,\dots,k,\ \text{and }t_m>\theta_{1,m}\text{ for some }m\}
\]
has positive Lebesgue measure, and on this set case iii) occurs. Hence again $\int_{\mathbf R^{k-2}} I(t_{3:k})\,dt_{3:k}<0.$ Therefore,
\[
\frac{\partial}{\partial \theta_{2,(1,2)}}\bigl(1-J^m(x,\theta_1,\theta_2)\bigr)<0
\qquad\text{for all }x\ge 0.
\]
Combining Step 1 and Step 2 proves the result.
\end{proof}
\begin{lemma}[Monotonicity of $J(x, \theta_1, \theta_2)$ in $\theta_2$]\label{lem:slepian-type}
For $\theta, \tilde{\theta} \in \Theta$ in $(\mathbf{R}^k_+\cup\mathbf{R}^k_-)\times \bar{\mathbf O}$, if it holds that
\begin{align*}
\tilde{\theta}_1= \theta_1 \text{ and }\tilde{\theta}_2 \geq \theta_2
\end{align*}
where the inequality holds element-wise, then $J^m$ in \eqref{def:J} satisfies that
\begin{align*}
J^m(x, {\theta}_1, {\theta}_2 ) \leq J^m(x, \tilde{\theta}_1, \tilde{\theta}_2)
\end{align*}
for any $x \in \mathbf{R}$. Furthermore, the inequality is strict if the two correlation matrices $\Tilde{\theta}_2$ and $\theta_2$ are positive definite and the strict inequality $\tilde{\theta}_{2,j}>{\theta}_{2,j}$ holds for some $j\in \{(a,b)\in[k]\times[k]:a<b\}$.
\end{lemma}
\begin{proof}
The result follows by applying the techniques in the proof of Theorem 2.1.1. in \cite{Tong1980probability} which is based on \cite{Slepian1962Bell}. I present the proof for completeness.
First suppose that both $\theta_2$ and $\tilde\theta_2$ are positive definite.
For $\lambda\in[0,1]$, define
\[
\bar\theta_1(\lambda):=\theta_1=\tilde\theta_1,
\qquad
\bar\theta_2(\lambda):=(1-\lambda)\theta_2+\lambda\tilde\theta_2.
\]
Since $\theta_2$ and $\tilde\theta_2$ are positive definite correlation
matrices, $\bar\theta_2(\lambda)$ is also a positive definite correlation
matrix for every $\lambda\in[0,1]$. By the chain rule,
\[
\frac{d}{d\lambda}J^m(x,\bar\theta_1(\lambda),\bar\theta_2(\lambda))
=
\sum_{j\in\{(a,b)\in[k]\times[k]:a<b\}}
\frac{\partial}{\partial \bar\theta_{2,j}(\lambda)}
J^m(x,\bar\theta_1(\lambda),\bar\theta_2(\lambda))
\cdot
(\tilde\theta_{2,j}-\theta_{2,j}).
\]
By Lemma \ref{lem:monotonedecreasing-in-theta2j}, $\frac{\partial}{\partial \bar\theta_{2,j}(\lambda)}
J^m(x,\bar\theta_1(\lambda),\bar\theta_2(\lambda))>0$ for every $j$, while by assumption $\tilde\theta_{2,j}-\theta_{2,j}\ge 0$ for every $j$. Hence
\[
\frac{d}{d\lambda} J^m(x,\bar\theta_1(\lambda),\bar\theta_2(\lambda))\ge 0
\qquad\text{for all }\lambda\in(0,1).
\]
Therefore the map $\lambda\mapsto J^m(x,\bar\theta_1(\lambda),\bar\theta_2(\lambda))$ is weakly increasing, and so
\[
J^m(x,\theta_1,\theta_2)
=
J^m(x,\bar\theta_1(0),\bar\theta_2(0))
\le
J^m(x,\bar\theta_1(1),\bar\theta_2(1))
=
J^m(x,\tilde\theta_1,\tilde\theta_2).
\]
If, in addition, $\tilde\theta_{2,j}>\theta_{2,j}$ for some $j$, then
\[
\frac{d}{d\lambda}J^m(x,\bar\theta_1(\lambda),\bar\theta_2(\lambda))>0
\qquad\text{for all }\lambda\in(0,1),
\]
and therefore $J^m(x,\theta_1,\theta_2)<J^m(x,\tilde\theta_1,\tilde\theta_2).$
Now suppose that at least one of $\theta_2$ and $\tilde\theta_2$ is singular.
For $\varepsilon>0$, define
\[
\theta_{2,\varepsilon}:=\frac{\theta_2+\varepsilon I_k}{1+\varepsilon},
\qquad
\tilde\theta_{2,\varepsilon}:=\frac{\tilde\theta_2+\varepsilon I_k}{1+\varepsilon}.
\]
Then $\theta_{2,\varepsilon}$ and $\tilde\theta_{2,\varepsilon}$ are positive
definite correlation matrices, and $\tilde\theta_{2,\varepsilon}\ge \theta_{2,\varepsilon}$ elementwise. Hence, by the positive definite case, $J^m(x,\theta_1,\theta_{2,\varepsilon})
\le
J^m(x,\theta_1,\tilde\theta_{2,\varepsilon})$ for every $\varepsilon>0.$ Since $J^m(x,\theta_1,\theta_2)$ is continuous in $\theta_2$, letting
$\varepsilon\downarrow 0$ yields
\[
J^m(x,\theta_1,\theta_2)\le J^m(x,\theta_1,\tilde\theta_2).
\]
This proves the weak inequality in general.
\end{proof}
\begin{lemma}\label{lemma:monotonicity-c}
Let $\theta_2,\tilde\theta_2\in \bar{\mathbf O}$ be such that
\[
\tilde\theta_2\ge \theta_2
\]
elementwise. Then for any $\alpha\in(0,1)$
\[
c^m(1-\alpha,\tilde\theta_2)\le c^m(1-\alpha,\theta_2).
\]
\end{lemma}
\begin{proof}
By Lemma \ref{lem:slepian-type}, for every
$\theta_1\in \mathbf R_+^k\cup \mathbf R_-^k$ and every $x\in \mathbf R$, $J^m(x,\theta_1,\theta_2)\le J^m(x,\theta_1,\tilde\theta_2).$ Therefore, for every $\theta_1\in \mathbf R_+^k\cup \mathbf R_-^k$, $J^{m,-1}(1-\alpha,\theta_1,\tilde\theta_2)
\le
J^{m,-1}(1-\alpha,\theta_1,\theta_2).$ Taking the supremum over $\theta_1\in \mathbf R_+^k\cup \mathbf R_-^k$ on both
sides yields
\[
\sup_{\theta_1\in \mathbf R_+^k\cup \mathbf R_-^k}
J^{m,-1}(1-\alpha,\theta_1,\tilde\theta_2)
\le
\sup_{\theta_1\in \mathbf R_+^k\cup \mathbf R_-^k}
J^{m,-1}(1-\alpha,\theta_1,\theta_2).
\]
By the definition of $c^m(1-\alpha,\theta_2)$, this is exactly $c^m(1-\alpha,\tilde\theta_2)\le c^m(1-\alpha,\theta_2).$
\end{proof}
\subsubsection{Proof of Theorem \ref{thm:LFtest-uniform-validity} with nonparametric bootstrap critical value}
\label{subsubsec:LF-nonparametric-bootstrap-validity}
We now prove Theorem \ref{thm:LFtest-uniform-validity} when the LF test uses the nonparametric bootstrap critical value in \eqref{def:LFC-bootstrap-critical-value}. Throughout this subsubsection, let $\hat P_n^*$ denote the conditional law of a nonparametric bootstrap sample $W_1^*,\ldots,W_n^*$ given the original sample. Let $\bar W_n^*$, $S_n^*$, and $\hat\Omega_n^*$ denote the bootstrap analogues of $\bar W_n$, $S_n$, and $\hat\Omega_n$, respectively. Conditional on the original sample, the bootstrap CDF in \eqref{def:LFC-bootstrap-critical-value} can be written as
\begin{align*}
J_n^\ell(x,\theta_1,\hat P_n)
=
\hat P_n^*\left\{
T^\ell\left(
\sqrt n\,S_n^{*-1}(\bar W_n^*-\bar W_n)
+S_n^{*-1}\theta_1,
\hat\Omega_n^*
\right)
\le x
\right\},
\end{align*}
where the second argument of $T^\ell$ is suppressed for $\ell=m,s$. For $\ell=q$, the regularization convention in Remark \ref{remark:regularized-correlation} is applied to the bootstrap correlation matrix as well. Bootstrap quantities may be defined arbitrarily on the event that some bootstrap marginal variance is zero; the argument below implies that the conditional probability of this event converges to zero.
The proof uses the bootstrap argument of \citet{RomanoShaikh2012AoS} for the studentized multivariate mean. We first state the sequential version needed here. The additional work relative to their Theorems~3.7--3.8 is that the nuisance value $\sqrt n\mu(P_n)$ may vary with $n$ and some of its standardized coordinates may diverge.
\begin{lemma}\label{lem:bootstrap-conv-to-Jn}
Consider a sequence $\{P_n\in\mathbf P:n\ge1\}$ and suppose \eqref{def:uniform-integrability} holds. Suppose that, for some nonempty set $I\subseteq[k]$, either
\begin{align}
\frac{\sqrt n\,\mu_j(P_n)}{\sigma_j(P_n)}&\to\delta_j\in(-\infty,0], &&j\in I,
&\frac{\sqrt n\,\mu_j(P_n)}{\sigma_j(P_n)}&\to-\infty, &&j\notin I,
\label{eq:bootstrap-drift-negative}
\end{align}
or the sign-reversed version of \eqref{eq:bootstrap-drift-negative} holds. Then, for $\ell\in\{m,s\}$,
\begin{align}
\sup_{x\in\mathbf R}
\left|
J_n^\ell(x,\sqrt n\,\mu(P_n),\hat P_n)
-
P_n\{T_n^\ell\le x\}
\right|
\overset{P_n}{\longrightarrow}0.
\label{eq:bootstrap-conv-to-Jn}
\end{align}
If, in addition,
$\inf_{P\in\mathbf P}\lambda_{\min}(\Omega(P))>0$, then \eqref{eq:bootstrap-conv-to-Jn} also holds for $\ell=q$.
\end{lemma}
\begin{proof}
It suffices to establish the result along an arbitrary subsequence. By compactness of $\bar{\mathbf O}$, pass to a further subsequence, without relabeling, such that
\begin{align}
\Omega(P_n)\to\Omega^*\in\bar{\mathbf O}.
\label{eq:bootstrap-Omega-limit}
\end{align}
For $\ell=q$, the uniform lower bound on the smallest eigenvalue implies $\Omega^*\succ0$.
We first derive the conditional bootstrap central limit theorem and studentization results. Let $\mathbf P'$ denote the set of distributions on $\mathbf R^k$ with finite and strictly positive marginal variances. For $(Q,P)\in\mathbf P'\times\mathbf P$, define, as in Lemma~S.12.1 of the supplement to \citet{RomanoShaikh2012AoS},
\begin{align*}
r_j(\lambda,Q)
&:=
\mathbb E_Q\left[
\left(\frac{W_j-\mu_j(Q)}{\sigma_j(Q)}\right)^2
I\left\{
\left|\frac{W_j-\mu_j(Q)}{\sigma_j(Q)}\right|>\lambda
\right\}
\right],\\
\rho(Q,P)
&:=
\max\left\{
\max_{1\le j\le k}
\int_0^\infty
|r_j(\lambda,Q)-r_j(\lambda,P)|e^{-\lambda}\,d\lambda,
\ \|\Omega(Q)-\Omega(P)\|_{\max}
\right\}.
\end{align*}
Consider first arbitrary deterministic sequences $Q_n\in\mathbf P'$ and $P_n\in\mathbf P$ satisfying
\begin{align}
\rho(Q_n,P_n)&\to0,
\label{eq:Qn-rho-close}\\
\max_{1\le j\le k}
\left|
\frac{\sigma_j(Q_n)}{\sigma_j(P_n)}-1
\right|&\to0.
\label{eq:Qn-scale-close}
\end{align}
The first paragraph of the proof of Lemma~S.12.1 of \citet{RomanoShaikh2012AoS} shows that \eqref{eq:Qn-rho-close}, together with the standardized uniform integrability of $P_n$, implies
\begin{align}
\lim_{\lambda\to\infty}\limsup_{n\to\infty}r_j(\lambda,Q_n)=0,
\qquad j=1,\ldots,k.
\label{eq:Qn-sequential-UI}
\end{align}
Thus the sequence $Q_n$ satisfies the same sequential standardized uniform integrability condition. We record explicitly that this is enough to apply Lemmas \ref{alem:consistency-sample-corr}--\ref{alem:uniformCLT} to the class $\{Q_n:n\ge1\}$. Because $r_j(\lambda,Q_n)$ is nonincreasing in $\lambda$, \eqref{eq:Qn-sequential-UI} implies that, for every $\varepsilon>0$, there exist $\lambda_0<\infty$ and $N<\infty$ such that
\[
\sup_{n\ge N}r_j(\lambda,Q_n)\le 2\varepsilon
\qquad\text{for all }\lambda\ge\lambda_0.
\]
For each of the finitely many $n<N$, finite second moments imply $r_j(\lambda,Q_n)\to0$ as $\lambda\to\infty$. Hence
\begin{align}
\lim_{\lambda\to\infty}
\sup_{n\ge1}r_j(\lambda,Q_n)=0,
\qquad j=1,\ldots,k,
\label{eq:Qn-class-UI}
\end{align}
so the class $\{Q_n:n\ge1\}$ satisfies \eqref{def:uniform-integrability}. Moreover, by the definition of $\rho$ and \eqref{eq:bootstrap-Omega-limit},
\begin{align}
\Omega(Q_n)\to\Omega^*.
\label{eq:Qn-Omega-limit}
\end{align}
Let $\bar W_n(Q_n)$, $S_n(Q_n)$, and $\hat\Omega_n(Q_n)$ denote the sample mean, diagonal matrix of sample standard deviations, and sample correlation matrix computed from an i.i.d. sample of size $n$ from $Q_n$, and write
\[
D(Q_n):=\operatorname{diag}\bigl(\sigma_1(Q_n),\ldots,\sigma_k(Q_n)\bigr).
\]
Lemma \ref{alem:uniformCLT}, applied to the class $\{Q_n:n\ge1\}$ and the standardized variables $D(Q_n)^{-1}\{W_i-\mu(Q_n)\}$, together with \eqref{eq:Qn-Omega-limit}, gives
\begin{align}
D(Q_n)^{-1}\sqrt n\{\bar W_n(Q_n)-\mu(Q_n)\}
\Rightarrow N(0_k,\Omega^*)
\qquad\text{under }Q_n.
\label{eq:Qn-unstudentized-CLT}
\end{align}
By Lemma \ref{alem:consistency-sample-variance}, applied coordinatewise to the class $\{Q_n:n\ge1\}$,
\begin{align}
S_n(Q_n)^{-1}D(Q_n)
\overset{Q_n}{\longrightarrow}I_k.
\label{eq:Qn-own-scale}
\end{align}
Combining \eqref{eq:Qn-own-scale} with \eqref{eq:Qn-scale-close} yields
\begin{align}
S_n(Q_n)^{-1}D(P_n)
&=
S_n(Q_n)^{-1}D(Q_n)D(Q_n)^{-1}D(P_n)
\overset{Q_n}{\longrightarrow}I_k.
\label{eq:Qn-Pn-scale}
\end{align}
Therefore, by Slutsky's theorem,
\begin{align}
\sqrt n\,S_n(Q_n)^{-1}\{\bar W_n(Q_n)-\mu(Q_n)\}
\Rightarrow N(0_k,\Omega^*)
\qquad\text{under }Q_n.
\label{eq:Qn-studentized-CLT}
\end{align}
Finally, Lemma \ref{alem:consistency-sample-corr}, applied to the class $\{Q_n:n\ge1\}$, together with \eqref{eq:Qn-Omega-limit}, gives
\begin{align}
\hat\Omega_n(Q_n)
\overset{Q_n}{\longrightarrow}\Omega^*.
\label{eq:Qn-corr-consistency}
\end{align}
We now replace $Q_n$ by the empirical distribution $\hat P_n$. First, Lemma \ref{alem:consistency-sample-variance} and $\sigma_j(P_n)>0$ give
\begin{align}
P_n\{\min_{1\le j\le k}S_{j,n}>0\}\to1.
\label{eq:Phat-positive-variances}
\end{align}
Hence $\hat P_n\in\mathbf P'$ with probability approaching one, so $\rho(\hat P_n,P_n)$ is well defined on an event whose probability tends to one; on the complementary event, it may be defined arbitrarily. Lemma~S.12.2 of the supplement to \citet{RomanoShaikh2012AoS} establishes
\[
\int_0^\infty
|r_j(\lambda,\hat P_n)-r_j(\lambda,P_n)|e^{-\lambda}\,d\lambda
\overset{P_n}{\longrightarrow}0,
\qquad j=1,\ldots,k,
\]
and Lemma \ref{alem:consistency-sample-corr}, which is Lemma~S.7.1 of the same supplement, gives
$\|\Omega(\hat P_n)-\Omega(P_n)\|_{\max}\overset{P_n}{\to}0$.
Consequently,
\begin{align}
\rho(\hat P_n,P_n)\overset{P_n}{\longrightarrow}0.
\label{eq:Phat-rho-close}
\end{align}
Furthermore, since $\sigma_j(\hat P_n)=S_{j,n}$ on the event in \eqref{eq:Phat-positive-variances}, Lemma \ref{alem:consistency-sample-variance}, which is Lemma~S.6.1 of the supplement to \citet{RomanoShaikh2012AoS}, implies
\begin{align}
\max_{1\le j\le k}
\left|
\frac{\sigma_j(\hat P_n)}{\sigma_j(P_n)}-1
\right|
=\max_{1\le j\le k}
\left|
\frac{S_{j,n}}{\sigma_j(P_n)}-1
\right|
\overset{P_n}{\longrightarrow}0.
\label{eq:Phat-scale-close}
\end{align}
To pass from the deterministic $Q_n$ argument to the conditional bootstrap law, consider an arbitrary subsequence. By \eqref{eq:Phat-positive-variances}--\eqref{eq:Phat-scale-close}, there is a further subsequence along which \eqref{eq:Phat-rho-close} and \eqref{eq:Phat-scale-close} hold almost surely and $\hat P_n\in\mathbf P'$ eventually almost surely. For almost every realization of the original sample along that further subsequence, $Q_n=\hat P_n$ is therefore eventually a deterministic sequence satisfying \eqref{eq:Qn-rho-close}--\eqref{eq:Qn-scale-close}. Applying \eqref{eq:Qn-Pn-scale}--\eqref{eq:Qn-corr-consistency} pathwise and using
\[
\mu(\hat P_n)=\bar W_n,
\qquad
\bar W_n(\hat P_n)=\bar W_n^*,
\qquad
S_n(\hat P_n)=S_n^*,
\qquad
\hat\Omega_n(\hat P_n)=\hat\Omega_n^*,
\]
gives
\begin{align}
\sqrt n\,S_n^{*-1}(\bar W_n^*-\bar W_n)
&\rightsquigarrow_{\hat P_n^*}N(0_k,\Omega^*),
\label{eq:bootstrap-centered-clt}\\
S_n^{*-1}D(P_n)
&\overset{\hat P_n^*}{\longrightarrow}I_k,
\qquad
\hat\Omega_n^*
\overset{\hat P_n^*}{\longrightarrow}\Omega^*,
\label{eq:bootstrap-studentization}
\end{align}
in $P_n$ probability where $\rightsquigarrow_{\hat P_n^*}$ denotes conditional weak convergence under the bootstrap law. Since the original subsequence was arbitrary, \eqref{eq:bootstrap-centered-clt}--\eqref{eq:bootstrap-studentization} hold along the full sequence. This is the same deterministic-sequence-to-empirical-distribution device used in Theorem~2.4 and in the proofs of Theorems~3.7--3.8 of \citet{RomanoShaikh2012AoS}.
For comparison, the three lemmas stated above give for the original sample
\begin{align}
\sqrt n\,S_n^{-1}(\bar W_n-\mu(P_n))
&\Rightarrow N(0_k,\Omega^*),
\label{eq:sample-centered-clt-bootstrap-proof}\\
S_n^{-1}D(P_n)
&\overset{P_n}{\longrightarrow}I_k,
\qquad
\hat\Omega_n
\overset{P_n}{\longrightarrow}\Omega^*.
\label{eq:sample-studentization-bootstrap-proof}
\end{align}
We now incorporate the drifting nuisance sequence directly through $\sqrt n\,\mu(P_n)$. We prove the result under \eqref{eq:bootstrap-drift-negative}; the sign-reversed case is identical. By \eqref{eq:bootstrap-studentization}, for each $j\in I$,
\begin{align*}
\frac{\sqrt n\,\mu_j(P_n)}{S_{j,n}^*}
=
\frac{\sigma_j(P_n)}{S_{j,n}^*}
\frac{\sqrt n\,\mu_j(P_n)}{\sigma_j(P_n)}
\overset{\hat P_n^*}{\longrightarrow}\delta_j
\end{align*}
in $P_n$-probability, whereas the same expression converges to $-\infty$ conditionally for $j\notin I$. Equations \eqref{eq:sample-studentization-bootstrap-proof} give the corresponding limits for $\sqrt n\,\mu_j(P_n)/S_{j,n}$. Combining these results with \eqref{eq:bootstrap-centered-clt} and \eqref{eq:sample-centered-clt-bootstrap-proof} shows that the bootstrap and sampling studentized vectors have the same finite-coordinate weak limits and the same divergent-coordinate behavior.
The corresponding limits of the test statistics have already been derived
in the proof of Lemma \ref{lem:conv-ip-to-Jn}. In particular, if
$I=[k]$, the common limit is
\begin{align}
V_I^\ell
:=
T^\ell(Z+\delta,\Omega^*),
\qquad
Z\sim N(0_k,\Omega^*),
\label{eq:bootstrap-limit-full-I}
\end{align}
where the second argument is suppressed for $\ell=m,s$. If
$I\subsetneq[k]$, the common limits are
\begin{align}
V_I^m
&:=
\max_{j\in I}(Z_j+\delta_j),
\label{eq:bootstrap-limit-m}\\
V_I^s
&:=
\sum_{j\in I}
(Z_j+\delta_j)^2 I\{Z_j+\delta_j\ge0\},
\label{eq:bootstrap-limit-s}
\end{align}
where
$Z_I\sim N(0_{|I|},\Omega^*_{I\times I})$. Under the uniform
eigenvalue condition, the corresponding limit for $\ell=q$ is
\begin{align}
V_I^q
:=
\inf_{\nu\in\mathbf R_-^{|I|}}
(Z_I+\delta_I-\nu)'
(\Omega^*_{I\times I})^{-1}
(Z_I+\delta_I-\nu).
\label{eq:bootstrap-limit-q}
\end{align}
For $\ell=q$, this is the same partial-divergence reduction established
in the proof of Lemma \ref{lem:conv-ip-to-Jn}; the uniform eigenvalue
condition ensures $\Omega^*\succ0$ and makes the regularization in
Remark \ref{remark:regularized-correlation} asymptotically irrelevant.
Let $F_I^\ell$ denote the CDF of $V_I^\ell$. The proof of
Lemma \ref{lem:conv-ip-to-Jn} also establishes that
\begin{align}
\sup_{x\in\mathbf R}
\left|
P_n\{T_n^\ell\le x\}-F_I^\ell(x)
\right|
\to0.
\label{eq:sampling-to-common-limit-bootstrap-proof}
\end{align}
For $\ell=m$, this follows from continuity of the limiting CDF and
P\'olya's theorem. For $\ell\in\{s,q\}$, the limiting CDF may have an
atom at zero, and the proof of Lemma \ref{lem:conv-ip-to-Jn} establishes
uniform convergence by separately verifying convergence at zero and
then applying the usual P\'olya argument on $(0,\infty)$.
The same argument applies to the conditional bootstrap distribution.
Indeed, \eqref{eq:bootstrap-centered-clt} and
\eqref{eq:bootstrap-studentization} give the conditional analogues of
the centered CLT and studentization results used in the proof of
Lemma \ref{lem:conv-ip-to-Jn}. Moreover,
\[
\frac{\sqrt n\,\mu_j(P_n)}{S_{j,n}^*}
\overset{\hat P_n^*}{\longrightarrow}\delta_j
\quad\text{for }j\in I,
\qquad
\frac{\sqrt n\,\mu_j(P_n)}{S_{j,n}^*}
\overset{\hat P_n^*}{\longrightarrow}-\infty
\quad\text{for }j\notin I,
\]
in $P_n$-probability. Therefore, repeating the deterministic
partial-divergence and split-at-zero arguments from the proof of
Lemma \ref{lem:conv-ip-to-Jn} conditionally yields
\begin{align}
\sup_{x\in\mathbf R}
\left|
J_n^\ell(x,\sqrt n\,\mu(P_n),\hat P_n)
-
F_I^\ell(x)
\right|
\overset{P_n}{\longrightarrow}0.
\label{eq:bootstrap-to-common-limit-bootstrap-proof}
\end{align}
Combining
\eqref{eq:sampling-to-common-limit-bootstrap-proof} and
\eqref{eq:bootstrap-to-common-limit-bootstrap-proof} by the triangle
inequality proves \eqref{eq:bootstrap-conv-to-Jn}.
\end{proof}
We now prove uniform asymptotic validity when
\[
\hat c_n^\ell=c_n^{\ell*}(1-\alpha,\hat P_n).
\]
Suppose, by way of contradiction, that the conclusion of Theorem
\ref{thm:LFtest-uniform-validity} fails. Then there exist $\eta>0$, a
subsequence $n_l$, and $P_{n_l}\in\mathbf P_0$ such that
\begin{align}
P_{n_l}\left\{
T_{n_l}^\ell>
c_{n_l}^{\ell*}(1-\alpha,\hat P_{n_l})
\right\}
>
\alpha+\eta
\qquad\text{for every }l.
\label{eq:bootstrap-validity-contradiction}
\end{align}
The subsequence reduction is identical to that in the preceding proof for the
parametric bootstrap critical value. Thus, after passing to further subsequences
if necessary and using sign symmetry, we may assume
$\mu(P_{n_l})\in\mathbf R_-^k$ for every $l$,
\[
\Omega(P_{n_l})\to\Omega^*\in\bar{\mathbf O},
\]
and each standardized mean
\[
\frac{\sqrt{n_l}\mu_j(P_{n_l})}{\sigma_j(P_{n_l})},
\qquad j\in[k],
\]
has an extended limit in $[-\infty,0]$. For $\ell=q$, the uniform eigenvalue
condition implies $\Omega^*\succ0$. As in the preceding proof, there are two
cases.
\medskip
\noindent
\textbf{Case 1: all standardized means diverge to $-\infty$.}
Suppose
\[
\frac{\sqrt{n_l}\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}\to-\infty
\qquad\text{for every }j\in[k].
\]
By Lemma \ref{lem:statistic-degenerate-case}, $T_{n_l}^m\overset{P_{n_l}}{\longrightarrow}-\infty,$ $T_{n_l}^s\overset{P_{n_l}}{\longrightarrow}0,$ and, under the uniform eigenvalue condition, $T_{n_l}^q\overset{P_{n_l}}{\to}0$.
It remains only to verify that the nonparametric bootstrap LF critical value
has the same qualitative lower bound used in Case 1 of the preceding
parametric-bootstrap proof. For every fixed $a\in\mathbf R^k$, equations
\eqref{eq:bootstrap-centered-clt}--\eqref{eq:bootstrap-studentization} imply,
at every continuity point $x$ of the limiting CDF,
\begin{align}
J_{n_l}^\ell
(x,D(P_{n_l})a,\hat P_{n_l})
\overset{P_{n_l}}{\longrightarrow}
J^\ell(x,a,\Omega^*),
\label{eq:bootstrap-fixed-shift-convergence}
\end{align}
where the second argument of $T^\ell$ is suppressed for $\ell=m,s$.
For $\ell=m$, the lower-bound argument in Case 1 of the preceding proof
implies that there exists a finite $b\in\mathbf R$ such that
\[
J^m(b,0_k,\Omega^*)<1-\alpha.
\]
Since $0_k$ belongs to the LF parameter space,
\eqref{eq:bootstrap-fixed-shift-convergence} yields
\[
P_{n_l}\{
c_{n_l}^{m*}(1-\alpha,\hat P_{n_l})\le b
\}\to0.
\]
Together with $T_{n_l}^m\to-\infty$ in probability, this gives
\[
P_{n_l}\{
T_{n_l}^m>
c_{n_l}^{m*}(1-\alpha,\hat P_{n_l})
\}\to0.
\]
For $\ell=s$, and for $\ell=q$ under the uniform eigenvalue condition, the
corresponding argument in Case 1 of the preceding proof shows that there exist
$t_\ell>0$ and a finite $M_\ell>0$ such that, with $a_\ell:=(-M_\ell,0,\ldots,0)',$
\[
J^\ell(t_\ell,a_\ell,\Omega^*)<1-\alpha.
\]
Because $D(P_{n_l})a_\ell\in\mathbf R_-^k$, it is an admissible nuisance value
in the bootstrap LF supremum. Hence
\eqref{eq:bootstrap-fixed-shift-convergence} gives
\[
P_{n_l}\{
c_{n_l}^{\ell*}(1-\alpha,\hat P_{n_l})\le t_\ell
\}\to0.
\]
Since $T_{n_l}^\ell\to0$ in probability and $t_\ell>0$,
\[
P_{n_l}\{
T_{n_l}^\ell>
c_{n_l}^{\ell*}(1-\alpha,\hat P_{n_l})
\}\to0.
\]
Thus Case 1 contradicts \eqref{eq:bootstrap-validity-contradiction}.
\medskip
\noindent
\textbf{Case 2: at least one standardized mean has a finite limit.}
There exists a nonempty set $I\subseteq[k]$ such that
\[
\frac{\sqrt{n_l}\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}
\to\delta_j\in(-\infty,0]
\quad\text{for }j\in I,
\qquad
\frac{\sqrt{n_l}\mu_j(P_{n_l})}{\sigma_j(P_{n_l})}
\to-\infty
\quad\text{for }j\notin I.
\]
This case is the bootstrap analogue of Case 2 in the preceding proof. The only
change is that Lemma \ref{lem:conv-ip-to-Jn}, which approximates the sampling
CDF by its Gaussian analogue, is replaced by Lemma
\ref{lem:bootstrap-conv-to-Jn}. In particular,
\begin{align}
\sup_{x\in\mathbf R}
\left|
J_{n_l}^\ell
\left(x,\sqrt{n_l}\mu(P_{n_l}),\hat P_{n_l}\right)
-
P_{n_l}\{T_{n_l}^\ell\le x\}
\right|
\overset{P_{n_l}}{\longrightarrow}0.
\label{eq:bootstrap-case2-cdf-close}
\end{align}
Moreover, because
$\sqrt{n_l}\mu(P_{n_l})\in\mathbf R_-^k$, the definition of the
nonparametric bootstrap LF critical value gives
\begin{align}
c_{n_l}^{\ell*}(1-\alpha,\hat P_{n_l})
\ge
(J_{n_l}^\ell)^{-1}
\left(
1-\alpha,
\sqrt{n_l}\mu(P_{n_l}),
\hat P_{n_l}
\right).
\label{eq:bootstrap-LF-dominates-oracle-quantile}
\end{align}
Fix $\varepsilon>0$, and let
\[
\delta_l
:=
P_{n_l}\left\{
\sup_{x\in\mathbf R}
\left|
J_{n_l}^\ell
\left(x,\sqrt{n_l}\mu(P_{n_l}),\hat P_{n_l}\right)
-
P_{n_l}\{T_{n_l}^\ell\le x\}
\right|
>\varepsilon
\right\}.
\]
By Lemma \ref{lem:bootstrap-conv-to-Jn}, $\delta_l\to0$. Part (vi) of
Lemma A.1 of \citet{RomanoShaikh2012AoS}, applied to the sampling CDF of
$T_{n_l}^\ell$ and the random bootstrap CDF, therefore implies
\[
P_{n_l}\left\{
T_{n_l}^\ell>
(J_{n_l}^\ell)^{-1}
\left(
1-\alpha,
\sqrt{n_l}\mu(P_{n_l}),
\hat P_{n_l}
\right)
\right\}
\le
\alpha+\varepsilon+\delta_l.
\]
Combining this inequality with
\eqref{eq:bootstrap-LF-dominates-oracle-quantile} gives
\[
\limsup_{l\to\infty}
P_{n_l}\left\{
T_{n_l}^\ell>
c_{n_l}^{\ell*}(1-\alpha,\hat P_{n_l})
\right\}
\le
\alpha+\varepsilon.
\]
Since $\varepsilon>0$ is arbitrary, the limit superior is at most $\alpha$,
contradicting \eqref{eq:bootstrap-validity-contradiction}.
The case $\mu(P_{n_l})\in\mathbf R_+^k$ follows by reversing signs. Therefore,
\[
\limsup_{n\to\infty}
\sup_{P\in\mathbf P_0}
P\left\{
T_n^\ell>
c_n^{\ell*}(1-\alpha,\hat P_n)
\right\}
\le\alpha
\]
for $\ell\in\{m,s\}$, and also for $\ell=q$ under
$\inf_{P\in\mathbf P}\lambda_{\min}(\Omega(P))>0$.
This proves Theorem \ref{thm:LFtest-uniform-validity} for the
nonparametric bootstrap critical value.
\subsection{Lemmas for the conditional test}
\begin{lemma}\label{lem:branchwise-conditional-approx}
Consider a sequence $\{P_n\in \mathbf{P}: n \geq 1\}$ where $\mathbf{P}$ is a set of distributions on $\mathbf{R}^k$ satisfying \eqref{def:uniform-integrability}. Let $W_i$, $i=1,2,...,n$, be an i.i.d. sequence of random vectors with distribution $P_n$. Suppose for some non-empty set $I\subset \{1,2, ..., k\}$
\begin{align}\label{eq:neg-branch-regime}
\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \delta_j \in(-\infty, 0] \text{ for } j\in I\text{ and }\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to -\infty \text{ for }j \not\in I
\end{align}
or
\begin{align}\label{eq:pos-branch-regime}
\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \delta_j \in [0,\infty) \text{ for } j\in I\text{ and }\frac{\sqrt{n} \mu_j(P_n)}{ \sigma_j(P_n)} \to \infty \text{ for }j \not\in I.
\end{align}
Let $\gamma \in (0,\tfrac12)$ and $\ell\in\{m,s\}$. Then,
\begin{align*}
\limsup_{n\to\infty} P_n\{\psi_n^{\ell,-}(\gamma)=1\mid I_{\mathrm{neg}}=1\}\le \gamma
\quad\text{if}\quad
\liminf_{n\to\infty} P_n\{I_{\mathrm{neg}}=1\}>0
\quad \text{under \eqref{eq:neg-branch-regime},}\\
\limsup_{n\to\infty} P_n\{\psi_n^{\ell,+}(\gamma)=1\mid I_{\mathrm{pos}}=1\}\le \gamma
\quad\text{if}\quad
\liminf_{n\to\infty} P_n\{I_{\mathrm{pos}}=1\}>0
\quad \text{under \eqref{eq:pos-branch-regime}}.
\end{align*}
In addition, if $I=[k]$, then
\[
\limsup_{n\to\infty} P_n\{\psi_n^{\ell,0}(\gamma)=1\mid I_{\mathrm{origin}}=1\}\le \gamma
\quad\text{if}\quad
\liminf_{n\to\infty} P_n\{I_{\mathrm{origin}}=1\}>0.
\]
Furthermore, if the smallest eigenvalue of $\Omega(P)$ is bounded away from zero uniformly over $\mathbf{P}$,
$\inf_{P \in \mathbf{P}} \lambda_{\min}(\Omega(P))>0$, then the same conclusion holds for $\ell=q.$
\end{lemma}
\begin{proof}
We prove the negative-branch statement under \eqref{eq:neg-branch-regime}. The positive-branch statement follows by the same argument after reversing signs, and the origin branch can be shown similarly. To prove via contradiction, suppose that there exists a subsequence ${n_l}$ and $\eta>0$ such that
\begin{align}\label{eq:contra-neg-branch}
P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid I_{\mathrm{neg}}=1\}> \gamma+\eta \qquad \text{ for all }{n_l}.
\end{align}
By Bolzano-Weierstrass theorem there exists a further subsequence (which we still denote with $n_l$ for simplicity) $\Omega(P_{n_l})\to \Omega^\ast\in \bar{\mathbf O}$. By Lemma \ref{alem:consistency-sample-corr} and the triangle inequality, $\|\hat{\Omega}_n -\Omega^*\|\overset{P_{n_l}}{\to}0$. By continuity of $\kappa_\tau(\Omega)$,
\begin{align}\label{eq:kappa-convergence}
\hat{\kappa}_{n_l} := \kappa_\tau (\hat{\Omega}_{n_l}) \overset{P_{n_l}}{\to}\kappa_\tau ({\Omega}^*) =:\kappa^*
\end{align}
By Lemma \ref{alem:consistency-sample-variance} and by the techniques used in Lemma \ref{lem:statistic-degenerate-case}, $ \frac{\sqrt n\,\mu_j(P_{n_l})}{S_{j,{n_l}}} \overset{P_{n_l}}{\to}\delta_j \in (-\infty,0] $ for $j\in I$ and $\frac{\sqrt {n_l}\,\mu_j(P_{n_l})}{S_{j,{n_l}}}\overset{P_{n_l}}{\to}-\infty $ for $j\notin I$. Moreover, let
\begin{align*}
X_{n_l} := (X_{1,n_l},..., X_{k,n_l}) = \left(
\tfrac{\sqrt {n_l}\,\bar W_{1,{n_l}}}{S_{1,{n_l}}}, ..., \tfrac{\sqrt {n_l}\,\bar W_{k,{n_l}}}{S_{k,{n_l}}}
\right).
\end{align*}
By Lemma \ref{alem:uniformCLT} and Slutsky's theorem,
\begin{align}\label{eq:conditional-test-weak-convergence}
\left(
X_{j,n_l}
\right)_{j\in I}
\Rightarrow Y_I^\ast,
\qquad
Y_I^\ast\sim N(\delta_I,\Omega^\ast_{I\times I}),
\end{align}
where \(\delta_I=(\delta_j)_{j\in I}\).
Define a class of nonempty proper subsets of $[k]$ as
\[\mathcal A:=\{A\subset[k]:A\neq\varnothing,\ [k]\setminus A\neq\varnothing\}\]
and define the event indexed by $A \in \mathcal{A}$ as
\[
D_{n_l}^-(A):=\{\mathcal I_0=A,\ \mathcal I_-=[k]\setminus A\}.
\]
Since the events \(\{D_{n_l}^-(A): A\in\mathcal{A}\}\) are disjoint and partition \(\{I_{\mathrm{neg}}=1\}\), we have
\begin{align}\label{eq:conditional-test-decomposition}
P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid I_{\mathrm{neg}}=1\}
=
\textstyle\sum_{A\in\mathcal A}
P_{n_l}\{D_{n_l}^-(A)\mid I_{\mathrm{neg}}=1\}\,
P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D_{n_l}^-(A)\}
\end{align}
where $P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D_{n_l}^-(A)\}$ is set to zero if $P_{n_l}\{D_{n_l}^-(A)\}=0.$
Our goal is to show that the limit probability of $P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D_{n_l}^-(A)\}$ is bounded above by $\gamma$ for asymptotically relevant cases where $\lim_{n_l\to\infty}P_{n_l}\{D_{n_l}^-(A)\}>0$. If \(A\not\subset I\), then there exists \(j\in A\setminus I\) such that $X_{j,n_l}\overset{P_{n_l}}{\to}-\infty$, so
$P_{n_l}\{D_{n_l}^-(A)\} \leq P_{n_l}\{|X_{j,n_l}|\leq \hat\kappa_{n_{l}}\} \to0$, which implies $P_{n_l}\{D_{n_l}^-(A)|I_{\mathrm{neg}}=1\}\to0$. Hence only the cases with \(A\subset I\) are asymptotically relevant. For \(A\subset I\), the limit can be written as below:
\begin{align*}
\lim_{n_l\to\infty} P_{n_l}\{D_{n_l}^-(A)\}
&=
\lim_{n_l\to\infty} P_{n_l}\{
X_{j,n_l}<- \hat{\kappa}_{n_l} ~~ \forall j\in [k]\setminus A,~~
|X_{j,n_l}|\leq \hat{\kappa}_{n_l} ~~ \forall j\in A\}\\
&\overset{(a)}{=}
\lim_{ {n_l}\to\infty} P_{n_l}\{
X_{j,n_l}<-\hat{\kappa}_{n_l} ~~ \forall j\in I\setminus A,~~
|X_{j,n_l}|\leq \hat{\kappa}_{n_l} ~~ \forall j\in A\}\\
&\overset{(b)}{=} \lim_{ {n_l} \to\infty} P\{
Y_j^*<-\kappa ^*~~ \forall j\in I\setminus A,~~
|Y_j^* |\le \kappa^* ~~ \forall j\in A\}
=: D^{-}(A)
\end{align*}
where (a) holds because $P_{n_l}\{X_{j,n_l}<-\hat\kappa_{n_l}\} \to 1$ for $j \not\in I$ and (b) holds by \eqref{eq:kappa-convergence}-\eqref{eq:conditional-test-weak-convergence} and Slutsky's theorem. The limit probability $D^-(A)>0$ if $\Omega^*\succ0$ and it can be zero if $\Omega^*$ is not full-rank. If \(D^-(A)=0\), then \(P_{n_l}\{D_{n_l}^-(A)\}\to0\), so $P_{n_l}\{D_{n_l}^-(A)\mid I_{\mathrm{neg}}=1\}\to0.$ Thus this branch is asymptotically negligible in \eqref{eq:conditional-test-decomposition}.
It suffices to consider \(A\subset I\) such that \(D^-(A)>0\). For such $A$ and at the continuity point $t$ of the limit $N^{\ell,-}(\cdot;A)$, we have
\begin{align*}
&\lim_{n_l\to\infty} P_{n_l}\{
T_{n_l}^{\ell,-}\le t,\ D_{n_l}^-(A)\}\\
&\overset{(a)}{=}
\lim_{n_l\to\infty} P_{n_l}\{
T^{\ell,-}(X_{n_l}, \hat{\Omega}_{n_l}, A) \leq t,~
X_{j,n_l}<- \hat{\kappa}_{n_l} ~ \forall j\in I\setminus A,~
|X_{j,n_l}|\leq \hat{\kappa}_{n_l} ~ \forall j\in A\}\\
&\overset{(b)}{=} P\{
T^{\ell,-}(Y^*, \Omega^*, A) \leq t,~
Y_j^*<-\kappa ^*~~ \forall j\in I\setminus A,~~
|Y_j^* |\le \kappa^* ~~ \forall j\in A\}
=: N^{\ell, -}(t;A)
\end{align*}
where (a) holds because $\mathcal{I}_0=A$ on $ D_{n_l}^-(A)$ and $P_{n_l}\{X_{j,n_l}<-\hat\kappa_{n_l}\} \to 1$ for $j \not\in I$, and (b) holds by \eqref{eq:kappa-convergence}-\eqref{eq:conditional-test-weak-convergence} and because the map $x\mapsto T^{\ell,-}(x,\Omega,A)$ is continuous for $\ell=m,s$. For $\ell=q$, $(x,\Omega)\mapsto T^{q,-}(x,\Omega,A)$ is continuous in $\mathbf{R}\times \mathbf{O}$. Under $\inf_{P\in\mathbf P}\lambda_{\min}(\Omega(P))>0$, we have $\Omega^*\succ0$ and $\lambda_{\min}(\hat\Omega_{n_l})>\underline\lambda/2$ with probability approaching one for some $\underline\lambda>0$. Therefore (b) holds for $\ell=q$ as well.
Combining these two results, we obtain
\begin{align}\label{eq:neg-branch-pointwise-convergence}
F^{\ell,-}_{n_l}(t;A)
:=
\frac{
P_{n_l}\{T^{\ell,-}_{n_l}\le t,\ D^-_{n_l}(A)\}
}{
P_{n_l}\{D^-_{n_l}(A)\}
}
\to
\frac{N^{\ell,-}(t;A)}{D^-(A)}=:F^{\ell,-}(t;A)
\end{align}
for every continuity point \(t\) of \(F^{\ell,-}(\cdot;A)\). Let
\[
c_A^{\ell,-}
:=
\inf\{t\in\mathbf R:F^{\ell,-}(t;A)\ge 1-\gamma\}.
\]For \(\ell=m\), the distribution \(F^{m,-}(\cdot;A)\) is continuous on\(\mathbf R\), so at its quantile \(c_A^{m,-}\) as well. For \(\ell=s,q\), \(F^{\ell,-}(\cdot;A)\) is continuous on
\((0,\infty)\), but it may be discontinuous at zero. We therefore distinguish two cases for \(\ell=s,q\): \(c_A^{\ell,-}>0\) and \(c_A^{\ell,-}=0\). The argument used for \(c_A^{\ell,-}>0\) also covers the case \(\ell=m\).
\medskip \noindent
\textit{Case 1: Either \(\ell=m\), or \(\ell\in\{s,q\}\) and \(c_A^{\ell,-}>0\).} Let
\(
\delta_{n_l}
:=
(
\frac{\sqrt{n_l}\mu_1(P_{n_l})}{\sigma_1(P_{n_l})} \wedge0,
\ldots,
\frac{\sqrt{n_l}\mu_k(P_{n_l})}{\sigma_k(P_{n_l})} \wedge0
).
\)
Then $\delta_{n_l} \in \mathbf{R}^k_-$ and $\delta_{n_l}\to \delta$ in \eqref{eq:neg-branch-regime}. Let
\[
q_A^{\ell,-}(\delta_{n_l},\hat\Omega_{n_l})
:=
(L_A^{\ell,-})^{-1}(1-\gamma,\delta_{n_l},\hat\Omega_{n_l})
\]
be the Gaussian conditional \((1-\gamma)\)-quantile associated with the local sequence
\(\delta_{n_l}\). Repeating the same numerator-denominator argument used above for the
Gaussian conditional law gives
\begin{align}\label{eq:neg-branch-quantile-convergence}
q_A^{\ell,-}(\delta_{n_l},\hat\Omega_{n_l})
\overset{P_{n_l}}{\to}
c_A^{\ell,-}.
\end{align}
For any small \(\varepsilon>0\) such that $c^{\ell,-}_A-\varepsilon$ is a continuity point of $F^{\ell,-}(\cdot;A),$ we have
\begin{align*}
\limsup_{n_l\to \infty}P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D^-_{n_l}(A)\}
&\overset{(a)}{\leq}\limsup_{n_l\to \infty}
P_{n_l}\{
T^{\ell,-}_{n_l}>
q_A^{\ell,-}(\delta_{n_l},\hat\Omega_{n_l})
\mid D^-_{n_l}(A)
\} \\
&\overset{(b)}{\leq}\limsup_{n_l\to \infty}
P_{n_l}\{
T^{\ell,-}_{n_l}>c_A^{\ell,-}-\varepsilon
\mid D^-_{n_l}(A)
\}\\
&\overset{(c)}{=}1-F^{\ell,-}(c_A^{\ell,-}-\varepsilon ;A)
\end{align*}
where (a) holds because $q^{\ell, -}_A(\delta_{n_l}, \hat{\Omega}_{n_l})\leq c^{\ell,-}_{\mathcal{I}_0} (1-\gamma, \hat{\Omega}_{n_l})$ conditional on $D^{-}_{n_l} (A)$ by definition of the critical value; (b) holds
by \eqref{eq:neg-branch-quantile-convergence} and
\(P_{n_l}\{D^-_{n_l}(A)\}\to D^-(A)>0\); and (c) follows from \eqref{eq:neg-branch-pointwise-convergence}. Since $F^{\ell,-}$ can have at most countably many discontinuities, we can let $\varepsilon\to0$ along continuity points. Then,
\begin{align*}
\limsup_{n_l\to\infty}
P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D^-_{n_l}(A)\}
&\le
1-F^{\ell,-}(c_A^{\ell,-};A)
\le \gamma
\end{align*}
for $\ell\in \{m,s\}$ and $\gamma\in(0,\tfrac12)$. If $\inf_{P\in \mathbf{P}} \lambda_{\min}(\Omega(P))>0$, the same conclusion holds for $\ell=q$.
\medskip \noindent
\textit{Case 2: \(\ell\in\{s,q\}\) and \(c_A^{\ell,-}=0\). }Since $\{T^{\ell,-}(x,\Omega,A)=0\}
=
\{x_j\le0 \text{ for all } j\in A\},$ by the same numerator--denominator argument used above,
\begin{align}\label{eq:neg-branch-pointwise-conv-at-zero}
F_{n_l}^{\ell,-}(0;A)\to F^{\ell,-}(0;A).
\end{align}
Then, we have
\begin{align*}
&\limsup_{n_l\to\infty} P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D^-_{n_l}(A)\}
\overset{(a)}{\le}
\limsup_{n_l\to\infty}P_{n_l}\{T_{n_l}^{\ell,-}>0\mid D^-_{n_l}(A)\}
\overset{(b)}{=} 1-F^{\ell,-}(0;A) \overset{(c)}{\leq} \gamma
\end{align*}
where (a) holds because the mapping $x \mapsto T^{\ell,-}(x,A)$ is non-negative so $c^{\ell,-}_A(1-\gamma, \hat{\Omega}_{n_l})\geq0$; (b) holds by \eqref{eq:neg-branch-pointwise-conv-at-zero}; and (c) follows by the definition of the quantile as \(c_A^{\ell,-}=0\).
\medskip\noindent
Combining Case 1 and Case 2, for every asymptotically relevant \(A\subset I\), we have
$$\limsup_{n_l\to\infty} P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid D^-_{n_l}(A)\}\leq \gamma.$$
Since all other \(A\)'s are asymptotically negligible in \eqref{eq:conditional-test-decomposition} and $\mathcal{A}$ is finite, we obtain
\[
\limsup_{l\to\infty}
P_{n_l}\{\psi_{n_l}^{\ell,-}(\gamma)=1\mid I_{\mathrm{neg}}=1\}
\le \gamma
\]
which contradicts \eqref{eq:contra-neg-branch}.
\end{proof}
\begin{lemma}\label{lem:kappa-continuity}
Fix \(\tau\in(0,1/2)\). The map $\theta_2\mapsto\kappa_\tau(\theta_2)$ defined in \eqref{def:kappa-screening-threshold} is continuous on \(\bar{\mathbf O}\).
\end{lemma}
\begin{proof}
Let \(\theta_{2,n}\to\theta_2\), and write $X_n\sim N(0,\theta_{2,n})$, $X\sim N(0,\theta_2)$, $M_n:=\max_j X_{n,j}$ and $M:=\max_j X_j$. Convergence of the covariance matrices implies \(X_n\Rightarrow X\), and hence \(M_n\Rightarrow M\) by the continuous mapping theorem. Let \(F_{\theta_2}\) denote the CDF of \(M\). Since \(F_{\theta_2}\) is continuous, $F_{\theta_{2,n}}(x)\to F_{\theta_2}(x)$ for every $x\in\mathbf R$. Moreover, \(F_{\theta_2}\) is strictly increasing on \([0,\infty)\). Since $F_{\theta_2}(0)
=P(X_j\leq0\ \forall j)
\leq P(X_1\leq0)=\tfrac12<1-\tau,$ we have \(\kappa_\tau(\theta_2)>0\). Thus, for every sufficiently small \(\varepsilon>0\),
continuity and strict monotonicity give $F_{\theta_2}(\kappa_\tau(\theta_2)-\varepsilon)<1-\tau
<
F_{\theta_2}(\kappa_\tau(\theta_2)+\varepsilon).$
The pointwise convergence above implies that, for all sufficiently large
\(n\), $F_{\theta_{2,n}}(\kappa_\tau(\theta_2)-\varepsilon)< 1-\tau
<
F_{\theta_{2,n}}(\kappa_\tau(\theta_2)+\varepsilon).$ By the definition of the left quantile, $\kappa_\tau(\theta_2)-\varepsilon
<
\kappa_\tau(\theta_{2,n})
\leq \kappa_\tau(\theta_2)+\varepsilon.$
Hence
\(\kappa_\tau(\theta_{2,n})\to\kappa_\tau(\theta_2)\), proving the result.
\end{proof}
For $d\ge2$, $\kappa>0$, $t\in\mathbf R$, and
$\mu\in\mathbf R_+^d$, define
\begin{align}\label{def:Lzero-selective-general}
\mathcal L_{d,\kappa}(t,\mu)
:=
P\left\{
T_d^m(Z_\mu)\le t
\ \middle|\
|Z_{\mu,j}|\le\kappa,\ j=1,\ldots,d
\right\},
\qquad
Z_\mu\sim N(\mu,I_d),
\end{align}
where $T_d^m(x)
:=
\min\{
\max_{1\le j\le d}x_j,\
\max_{1\le j\le d}(-x_j)
\}.$
\begin{lemma}
\label{lem:k2-origin-schur}
Fix $\kappa>0$ and $t\in[0,\kappa)$. For every $s\ge0$, the map $r\mapsto
\mathcal L_{2,\kappa}\bigl(t,(s-r,r)\bigr)$ in \eqref{def:Lzero-selective-general} is weakly increasing on $[0,s/2]$. Consequently, $\mu\mapsto\mathcal L_{2,\kappa}(t,\mu)$ is Schur-concave on $\mathbf R_+^2$; that is, for any $\mu,\widetilde\mu\in\mathbf R_+^2$ such that $\mu_1+\mu_2=\widetilde\mu_1+\widetilde\mu_2$ and $\max\{\mu_1,\mu_2\}\geq \max\{\tilde\mu_1,\tilde\mu_2\}$, $\mathcal L_{2,\kappa}(t,\mu)
\le
\mathcal L_{2,\kappa}(t,\widetilde\mu)$.
\end{lemma}
\begin{proof}
We start by transforming the normal vector into $(V_1, V_2)$:
\[
V=(V_1, V_2):= (\tfrac{Z_{1}+Z_{2}}{2},~
\tfrac{Z_{1}-Z_{2}}{2})
\sim N((a,\delta), \tfrac12 I_2).
\]
where $a= s/2$ and $\delta=s/2-r \in[0,a].$ The major advantage of this transformation is that we can write the conditional CDF in terms of $V_2$:
\begin{align*}
1-\mathcal L_{2,\kappa}(t,(s-r,r))
=P\{ |V_2| > t+ |V_1| \mid |V_1|+ |V_2|\leq \kappa\}=:\bar{\mathcal{L}}_{2,\kappa}(t,(a,\delta)),
\end{align*}
where it follows from
\begin{gather*}
\{T^{m}_2(Z)>t\}
=\{|V_2|>t+|V_1|\},\quad
\{|Z_{1}|\le \kappa,\ |Z_{2}|\le \kappa\}
=\{|V_1|+|V_2|\le \kappa\}.
\end{gather*}
Using the change of variable $v_2 = -v_2'$, write $\bar{\mathcal{L}}_{2,\kappa}(t,(a,\delta))$ as
\begin{align*}
\bar{\mathcal{L}}_{2,\kappa}(t,(a,\delta))
=
\frac{
\iint_{\{|v_1|+|v_2|\le\kappa,\ |v_2|>t+|v_1|\}}
e^{-v_1^2-v_2^2+2av_1+2\delta v_2}\,dv_1dv_2
}{
\iint_{\{|v_1|+|v_2|\le\kappa\}}
e^{-v_1^2-v_2^2+2av_1+2\delta v_2}\,dv_1dv_2
}
=
\frac{
\int_0^{\kappa} n_a(v_2)\cosh(2\delta v_2)\,dv_2
}{
\int_0^{\kappa} d_a(v_2)\cosh(2\delta v_2)\,dv_2
},
\end{align*}
where $\cosh(2\delta v_2) = (e^{2\delta v_2} +e^{-2\delta v_2})/2$ and
\begin{align*}
d_a(v_2)
:=
2e^{-v_2^2}\int_{-(\kappa-v_2)}^{\kappa-v_2} e^{-v_1^2+2av_1}\,dv_1,
\quad
n_a(v_2)
:=
2e^{-v_2^2} I\{v_2>t\}\int_{-m(v_2)}^{m(v_2)} e^{-v_1^2+2av_1}\,dv_1,
\end{align*}
with $m(v_2):= \min\{v_2-t,\kappa-v_2\}$. Furthermore, write
\begin{align}
\bar{\mathcal{L}}_{2,\kappa}(t,(a,\delta))
=
\int_0^{\kappa} g_a(v_2)\,\nu_\delta(dv_2),
\label{eq:Ka-delta}
\end{align}
by defining
\[
g_a(v_2):= \frac{n_a(v_2)}{d_a(v_2)}
\quad \text{for }v_2\in[0,\kappa), \qquad \nu_\delta(dv_2)
=
\frac{d_a(v_2)\cosh(2\delta v_2)\,dv_2}
{\int_0^{\kappa} d_a(u)\cosh(2\delta u)\,du}.
\]
We show that the family \(\{\nu_\delta:\delta\ge 0\}\) is stochastically
increasing in \(\delta\). Fix \(0\le \delta_1<\delta_2\). Then
\[
\frac{d\nu_{\delta_2}}{d\nu_{\delta_1}}(v_2)
\propto
\frac{\cosh(2\delta_2 v_2)}{\cosh(2\delta_1 v_2)}.
\]
Its logarithmic derivative is positive for all $v_2>0$:
\[
\frac{d}{dv_2}
\log\frac{\cosh(2\delta_2 v_2)}{\cosh(2\delta_1 v_2)}
=
2\delta_2\tanh(2\delta_2 v_2)-2\delta_1\tanh(2\delta_1 v_2)>0
\]
because for any $v_2>0$ the map \(u\mapsto u\tanh(uv_2)\) is strictly increasing on
\([0,\infty)\) as
\[
\frac{d}{du}\{u\tanh(uv_2)\}
=
\tanh(uv_2)+uv_2\,\mathrm{sech}^2(uv_2)>0.
\]
Therefore \(d\nu_{\delta_2}/d\nu_{\delta_1}\) is increasing in \(v_2\). That is, the family
\(\{\nu_\delta:\delta\ge 0\}\) has monotone likelihood ratio in \(v_2\), and is
stochastically increasing in \(\delta\) by Theorem 1.C.1 in \cite{shaked2007stochastic-orders}.
We now show that \(g_a\) is nondecreasing on \([0,\kappa]\). First, if \(v_2\in[0,t]\), then \(n_a(v_2)=0\) and $g_a(v_2)=0.$ Second, if \(v_2\in[(\kappa+t)/2,\kappa]\), then \(n_a(v_2)=d_a(v_2)\) and so $g_a(v_2)=1.$ Finally, if \(v_2\in[t,(\kappa + t)/2]\), then \(m(v_2)=v_2-t\), and therefore
\[
g_a(v_2)
=
\frac{
\int_{-(v_2-t)}^{v_2-t} e^{-v_1^2+2av_1}\,dv_1
}{
\int_{-(\kappa-v_2)}^{\kappa-v_2} e^{-v_1^2+2av_1}\,dv_1
}.
\]
As \(v_2\) increases on \([t,(\kappa+t)/2]\), the numerator interval
\([-(v_2-t),v_2-t]\) expands, while the denominator interval
\([-(\kappa-v_2),\kappa-v_2]\) shrinks. Since the integrand
\(e^{-v_1^2+2av_1}\) is strictly positive, the numerator is increasing and the denominator is decreasing in \(v_2\). Hence \(g_a(v_2)\) is increasing on \([t,(\kappa+t)/2]\). Combining the three regions, \(g_a\) is nondecreasing on all of \([0,\kappa]\).
Since \(g_a\) is nondecreasing and \(\nu_\delta\) is stochastically increasing in
\(\delta\), $\bar{\mathcal{L}}_{2,\kappa}(t,(a,\delta))$ is increasing in \(\delta\). Consequently,
$\mathcal L_{2,\kappa}(t,(s-r,r))=1- \bar{\mathcal{L}}_{2,\kappa}(t,(a,\delta))$ is increasing in \(r\). Finally, $\mathcal L_{2,\kappa}(t,\cdot)$ is symmetric, and for every fixed coordinate sum it increases as the two coordinates become more balanced. Therefore it is Schur-concave on $\mathbf R_+^2$.
\end{proof}
\begin{lemma}\label{lem:origin-schur-k}
Fix an integer $d\ge2$, a constant $\kappa>0$, and
$t\in[0,\kappa)$. Then the map $\mu\mapsto\mathcal L_{d,\kappa}(t,\mu)
$ defined in \eqref{def:Lzero-selective-general} is Schur-concave on $\mathbf R_+^d$. In particular,
\begin{align}\label{eq:general-k-boundary-final}
\mathcal L_{d,\kappa}(t,\mu)
\geq
\mathcal L_{d,\kappa}
(
t,
(\textstyle\sum_{j=1}^d\mu_j,0,\ldots,0)'
)
\qquad
\forall\mu=(\mu_1,...,\mu_d)\in\mathbf R_+^d.\end{align}
\end{lemma}
\begin{proof}
We proceed by induction on $d$, holding $\kappa$ and $t$ fixed. For $d=2$, the conclusion follows from
Lemma \ref{lem:k2-origin-schur}. Suppose that $\mu\mapsto\mathcal L_{d,\kappa}(t,\mu)$ is Schur-concave on $\mathbf R_+^d$. We need to show that $\mu\mapsto\mathcal L_{d+1,\kappa}(t,\mu)$ is Schur-concave as well.
The conditional distribution can be written as below:
\begin{align*}
\mathcal L_{d,\kappa}(t,\mu)&= P\{ \max_{j}W_j\leq t \text{ or }\min_{j}W_j \geq -t\mid |W_j|\le \kappa ~ \forall j\in[k]\} \quad\text{for }W\sim N(\mu,I_k) \\
&=\textstyle \prod_{j=1}^d A(\mu_j)
+
\textstyle\prod_{j=1}^d B(\mu_j)
-
\textstyle\prod_{j=1}^d C(\mu_j)
\end{align*}
where for $Z_u\sim N(u,1)$
\begin{align*}
A(u)
&=
P\{Z_u \le t \mid |Z_u|\le \kappa\},~
B(u)
=
P\{Z_u\ge -t \mid |Z_u|\le \kappa\}, ~
C(u)
=
P\{|Z_u|\le t \mid |Z_u|\le \kappa\}.
\end{align*}
A direct algebraic rearrangement gives
\begin{align}
\begin{aligned}
\mathcal L_{d+1,\kappa}(t,\mu_1,\ldots,\mu_d,u)
&=
A(u)\,L_{d,\kappa}(t,\mu_1,\ldots,\mu_d)\\
&+
(B(u)-A(u))\textstyle\prod_{j=1}^d B(\mu_j)
+
(A(u)-C(u))\textstyle\prod_{j=1}^d C(\mu_j).
\end{aligned}
\label{eq:induction-identity}
\end{align}
We argue that \eqref{eq:induction-identity} is a nonnegative linear combination of Schur-concave functions of $\mu.$ Trivially, $A(u)\geq 0$ and $A(u)-C(u)= P\{Z+u <-t \mid |Z+u|\le \kappa\}\geq 0$. Also,
\begin{align*}
B(u)-A(u) &= P\{Z+u>t \mid |Z+u|\le \kappa\}-P\{Z+u<-t \mid |Z+u|\le \kappa\}
\end{align*}
is zero at $u=0$ and increasing in $u\geq0$ because $P\{Z+u>t \mid |Z+u|\le \kappa\}$ is increasing and $P\{Z+u<-t \mid |Z+u|\le \kappa\}$ is decreasing in $u$ by Lemma A.1 in \cite{lee2016AoS-post-selection-LASSO}. Furthermore, to see Schur-concavity of the mapping $\mu\mapsto \textstyle\prod_{j=1}^d B(\mu_j)$, consider
\begin{align*}
\textstyle\prod_{j=1}^d B(\mu_j)= \exp\{ \{\sum^d_{j=1} \log B(\mu_j)\}\}.
\end{align*}
By Lemma \ref{lem:conditional-interval-logconcavity} $\log B(\mu_j)$ is concave. Then, $\mu\mapsto\sum^d_{j=1} \log B(\mu_j)$ is symmetric and concave, so it is Schur-concave by Proposition C.2.f of \cite{MarshallOlkin1979Springer}. As the exponential map is strictly increasing, it preserves the Schur-concavity. If $t>0$, the same argument shows that $\textstyle\prod_{j=1}^d C(\mu_j)$ is also Schur-concave. If $t=0$, then $C(u)=0$ for every $u$, so this
product is identically zero and is trivially Schur-concave. It follows that, for each fixed $u\ge0$, $(\mu_1,\ldots,\mu_d)
\mapsto
\mathcal L_{d+1,\kappa}(t,\mu_1,\ldots,\mu_d,u)$ is Schur-concave.
It remains to deduce joint Schur-concavity of \eqref{eq:induction-identity} in all $d+1$ coordinates. Let
\(i\neq j\in\{1,\ldots,d+1\}\). We use Theorem 3.A.4 of \cite{MarshallOlkin1979Springer}, an equivalent representation of Schur-concavity: $L_{d+1,\kappa}(t,\mu_1,\ldots,\mu_d,\mu_{d+1}) $ is Schur-concave on $(0,\infty)^{d+1}$ if and only if
$$ (\mu_i - \mu_j) (\tfrac{\partial}{\partial \mu_i} \mathcal L_{d+1,\kappa}(t,\mu) - \tfrac{\partial}{\partial \mu_j} \mathcal L_{d+1,\kappa}(t,\mu)) \leq 0\quad \text{ for all }i\neq j.$$
Since \(d+1\ge 3\), choose an index
\(h\notin\{i,j\}\). By symmetry of \( \mathcal L_{d+1,\kappa}(t,\mu)\), we may permute coordinates so that
\(h\) becomes the \((d+1)\)-st coordinate and both \(i\) and \(j\) lie among the first \(d\) coordinates. For that permuted representation, the last coordinate is fixed,
and we have already shown that the function is Schur-concave in the first \(d\) coordinates. Thus the above pairwise inequality holds for the pair \((i,j)\). Since the pair was arbitrary, \(\mathcal L_{d+1,\kappa}(t,\mu)\) is Schur-concave on \(\mathbf R_+^{d+1}\). This completes the induction and proves \eqref{eq:general-k-boundary-final}.
\end{proof}
\begin{lemma}\label{lem:conditional-interval-logconcavity}
Fix \(\kappa>0\). For any interval \([a,b]\subseteq[-\kappa,\kappa]\) with \(a<b\), define
\[
G_{[a,b]}(u)
:=
\frac{\Phi(b-u)-\Phi(a-u)}
{\Phi(\kappa-u)-\Phi(-\kappa-u)}
=
P\{a\le W\le b\mid |W|\le \kappa\},
\qquad W\sim N(u,1).
\]
Then \(u\mapsto G_{[a,b]}(u)\) is log-concave on \(\mathbf R\).
\end{lemma}
\begin{proof}For a fixed interval \(I=[c,d]\), consider the conditional law of \(W\mid W\in I\sim N(u,1)\). Its density is
\[
f_{u,I}(w)
=
\frac{\phi(w-u) I\{w\in I\}}{\Phi(d-u)-\Phi(c-u)}
=
\exp\!\{uw-A_I(u)\}\phi(w) I\{w\in I\},
\]
where
\[
A_I(u)
:=
\frac{u^2}{2}+\log (\Phi(d-u)-\Phi(c-u)).
\]
Thus \(\{f_{u,I}:u\in\mathbf R\}\) is a one-parameter exponential family with
canonical statistic \(t(w)=w\). By Theorem 1.6.2 in \cite{bickel-doksum2015book},
\[
A_I''(u)= 1+ \frac{d^2}{du^2} \log(\Phi(d-u)-\Phi(c-u))=\mathbb{V}[W\mid W\in I].
\]
Since $\log G_{[a,b]}(u)=A_{[a,b]}(u)-A_{[-\kappa,\kappa]}(u)$, the identity above implies
\[
\frac{d^2}{du^2}\log G_{[a,b]}(u)
=
A_{[a,b]}''(u)-A_{[-\kappa,\kappa]}''(u)
=
\mathbb{V}[W\mid W\in [a,b]]-\mathbb{V}[W\mid W\in [-\kappa, \kappa]]\leq0
\]
where the inequality holds by Theorem 2 of \cite{chen2013partial} for any $u\in \mathbf R.$ This proves log-concavity.
\end{proof}
\subsection{Proof of Corollary \ref{cor:general-estimator-uniform-validity}}
\label{subsec:supporting-regular-estimators}
This subsection states the Gaussian approximation result and proves
Corollary~\ref{cor:general-estimator-uniform-validity}.
\begin{lemma}\label{lem:general-estimator-gaussian-limit}
Suppose Assumption \ref{ass:regular-asymptotically-linear} holds. Let
$\{P_n\}\subset\mathbf P$ satisfy
$\Omega(P_n)\to\Omega^*\in\bar{\mathbf O}$. Then
\begin{align}\label{eq:general-estimator-centered-limit}
\sqrt n\,\hat D_n^{-1}
\bigl(\hat\beta_n-\beta(P_n)\bigr)
\Rightarrow N(0_k,\Omega^*)
\end{align}
and $\hat\Omega_n\overset{P_n}{\to}\Omega^*$.
Moreover, let $I\subseteq[k]$ be nonempty and suppose either
\begin{align}\label{eq:general-estimator-positive-regime}
\frac{\sqrt n\,\beta_j(P_n)}{\sigma_j(P_n)}
&\to\delta_j\in[0,\infty),
&&j\in I,\nonumber\\
\frac{\sqrt n\,\beta_j(P_n)}{\sigma_j(P_n)}
&\to+\infty,
&&j\notin I,
\end{align}
or
\begin{align}\label{eq:general-estimator-negative-regime}
\frac{\sqrt n\,\beta_j(P_n)}{\sigma_j(P_n)}
&\to\delta_j\in(-\infty,0],
&&j\in I,\nonumber\\
\frac{\sqrt n\,\beta_j(P_n)}{\sigma_j(P_n)}
&\to-\infty,
&&j\notin I.
\end{align}
Then, under either regime,
\begin{align}\label{eq:general-estimator-partial-limit}
\left(\sqrt n\,\hat D_n^{-1}\hat\beta_n\right)_I
\Rightarrow
N(\delta_I,\Omega^*_{I\times I}).
\end{align}
In addition,
$\left(\sqrt n\,\hat D_n^{-1}\hat\beta_n\right)_j
\overset{P_n}{\to}+\infty$ for every $j\notin I$ under
\eqref{eq:general-estimator-positive-regime}, whereas
$\left(\sqrt n\,\hat D_n^{-1}\hat\beta_n\right)_j
\overset{P_n}{\to}-\infty$ for every $j\notin I$ under
\eqref{eq:general-estimator-negative-regime}.
\end{lemma}
\begin{proof}
For brevity in this proof, write
\[
X_n^\beta:=\sqrt n\,\hat D_n^{-1}\hat\beta_n.
\]
Write $\varphi_n:=\varphi_{P_n}$. Then
\[
\mathbb E_{P_n}[\varphi_n(U_i)]=0_k,
\qquad
\mathbb E_{P_n}[\varphi_n(U_i)\varphi_n(U_i)']
=\Omega(P_n)\to\Omega^*.
\]
For any fixed $a\in\mathbf R^k$, condition
\eqref{eq:UI-normalized-influence-function}, together with fixed $k$, implies
the Lindeberg condition for the triangular array
$\{a'\varphi_n(U_i)/\sqrt n:i=1,\ldots,n\}$. The triangular-array
Lindeberg--Feller theorem, in the form used in
\citet[Lemma~11.4.1]{LehmannRomano2005testing}, and the Cram\'er--Wold
device therefore give
\[
\frac{1}{\sqrt n}\sum_{i=1}^n\varphi_n(U_i)
\Rightarrow N(0_k,\Omega^*).
\]
Combining this result with \eqref{eq:uniform-asymptotic-linearity-general} and
\eqref{eq:studentization-general-consistency} proves
\eqref{eq:general-estimator-centered-limit}. The convergence of
$\hat\Omega_n$ follows from the second part of
\eqref{eq:studentization-general-consistency} and
$\Omega(P_n)\to\Omega^*$.
Next, define
\[
d_n:=\sqrt n\,D(P_n)^{-1}\beta(P_n).
\]
By \eqref{eq:studentization-general-consistency},
\[
\sqrt n\,\hat D_n^{-1}\beta(P_n)
=\hat D_n^{-1}D(P_n)d_n.
\]
Hence its coordinates have the limits stated in
\eqref{eq:general-estimator-positive-regime} or
\eqref{eq:general-estimator-negative-regime}. Since
\[
X_n^\beta
=
\sqrt n\,\hat D_n^{-1}
\bigl(\hat\beta_n-\beta(P_n)\bigr)
+
\sqrt n\,\hat D_n^{-1}\beta(P_n),
\]
\eqref{eq:general-estimator-partial-limit} follows from
\eqref{eq:general-estimator-centered-limit} and Slutsky's theorem.
\end{proof}
For brevity in this proof, write
\[
X_n^\beta:=\sqrt n\,\hat D_n^{-1}\hat\beta_n,
\qquad
d_n:=\sqrt n\,D(P_n)^{-1}\beta(P_n).
\]
Consider an arbitrary sequence
$\{P_n\}\subset\mathbf P_{\beta,0}$. From any subsequence, pass to a
further subsequence, still indexed by $n$, such that
$\Omega(P_n)\to\Omega^*\in\bar{\mathbf O}$, $\beta(P_n)$ belongs to the
same null orthant for every $n$, and every coordinate of $d_n$ has an
extended-real limit. By symmetry, it is enough to consider
$\beta(P_n)\in\mathbf R_+^k$.
Lemma \ref{lem:general-estimator-gaussian-limit} gives
\[
\sqrt n\,\hat D_n^{-1}\{\hat\beta_n-\beta(P_n)\}
\Rightarrow N(0_k,\Omega^*),
\qquad
\hat\Omega_n\overset{P_n}{\to}\Omega^*,
\]
and, whenever $I\subseteq[k]$ indexes the coordinates of $d_n$ having
finite limits,
\[
(X_n^\beta)_I\Rightarrow
N(\delta_I,\Omega^*_{I\times I}),
\qquad
X_{j,n}^\beta\overset{P_n}{\to}+\infty
\quad(j\notin I).
\]
These are the counterparts of the sample-mean convergence, studentization,
and partial-divergence results used in the proofs of Lemmas
\ref{lem:statistic-degenerate-case}, \ref{lem:conv-ip-to-Jn}, and
\ref{lem:branchwise-conditional-approx}.
We first consider the LF test. If $d_{j,n}\to+\infty$ for every $j\in[k]$,
the argument in Case 1 of the proof of Theorem
\ref{thm:LFtest-uniform-validity} applies after replacing
$\sqrt nS_n^{-1}\bar W_n$ by $X_n^\beta$. Thus the test statistic is
asymptotically degenerate in the null direction, while the same deterministic
lower bounds for the Gaussian LF critical value continue to apply, and hence
\[
\mathbb E_{P_n}[\phi_{n,\beta}^{\ell}]\to0.
\]
Suppose instead that the set $I$ of coordinates with finite limits is
nonempty. Define
\[
\hat d_n:=\sqrt n\,\hat D_n^{-1}\beta(P_n),
\]
\[
J_{n,\beta}^\ell(x)
:=P_n\{T^\ell(X_n^\beta,\hat\Omega_n)\le x\},
\qquad
\widetilde J_{n,\beta}^\ell(x)
:=J^\ell(x,\hat d_n,\hat\Omega_n).
\]
Applying the proof of Lemma \ref{lem:conv-ip-to-Jn}, with Lemma
\ref{lem:general-estimator-gaussian-limit} replacing the corresponding
sample-mean lemmas, gives
\[
\sup_{x\in\mathbf R}
|J_{n,\beta}^\ell(x)-\widetilde J_{n,\beta}^\ell(x)|
\overset{P_n}{\to}0
\]
for $\ell\in\{m,s\}$, and also for $\ell=q$ under
\eqref{eq:general-estimator-eigenvalue-condition}. This application includes
the same reduced-limit and atom-at-zero arguments used in that lemma. Since
$\hat d_n\in\mathbf R_+^k$, its Gaussian quantile is included in the
supremum defining $c^\ell(1-\alpha,\hat\Omega_n)$. Lemma~A.1(vi) of
\citet{RomanoShaikh2012AoS}, applied exactly as in the proof of Theorem
\ref{thm:LFtest-uniform-validity}, therefore yields
\[
\limsup_{n\to\infty}
\mathbb E_{P_n}[\phi_{n,\beta}^{\ell}]
\le\alpha.
\]
We next consider the conditional test. If all coordinates of $d_n$ diverge
to $+\infty$, then $P_n\{I_{\mathrm{null}}=1\}\to1$ and the rejection
probability converges to zero. Otherwise, the argument establishing the
opposite-direction screening bound in the proof of Theorem
\ref{thm:conditional-uniform-validity} applies with $X_n^\beta$ in place of
$\sqrt nS_n^{-1}\bar W_n$, and gives
\[
\limsup_{n\to\infty}
P_n\{\exists j\in[k]:X_{j,n}^\beta<
-\kappa_\tau(\hat\Omega_n)\}
\le\tau.
\]
Moreover, the proof of Lemma \ref{lem:branchwise-conditional-approx}
applies after the same replacement. Indeed, that proof uses only the
finite-coordinate Gaussian convergence, divergence of the remaining
coordinates, consistency of $\hat\Omega_n$ and
$\kappa_\tau(\hat\Omega_n)$, the zero Gaussian probability of the screening
boundaries, and inclusion of the relevant local drift in the least favorable
conditional critical value. All of these properties follow from Lemma
\ref{lem:general-estimator-gaussian-limit}, Assumption
\ref{ass:regular-asymptotically-linear}, and Lemma
\ref{lem:kappa-continuity}. The conditional quantile step again follows from
Lemma~A.1(vi) of \citet{RomanoShaikh2012AoS}. Consequently, with
$\gamma:=\alpha-\tau$, the conditional rejection probabilities in the
positive and origin cases are asymptotically bounded by $\gamma$ whenever
the corresponding conditioning event has asymptotically positive
probability; events whose probabilities vanish make a negligible
unconditional contribution.
The probability decomposition in the proof of Theorem
\ref{thm:conditional-uniform-validity} now gives
\[
\limsup_{n\to\infty}
\mathbb E_{P_n}[\psi_{n,\beta}^\ell]
\le\tau+\gamma=\alpha.
\]
The case $\beta(P_n)\in\mathbf R_-^k$ follows by reversing signs and
exchanging the positive and negative cases. Under
\eqref{eq:general-estimator-eigenvalue-condition}, the same arguments apply
to $\ell=q$ because every subsequential limit $\Omega^*$ is positive
definite. Since the original sequence and subsequence were arbitrary, the
subsequence principle proves the result.