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Testing Conditional Stochastic Dominance at Target Points
KEYWORDS: stochastic dominance, regression discontinuity design, induced order statistics, rank tests, permutation tests.
JEL classification codes: C12, C14.
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The concept of stochastic dominance has long been central to numerous areas of applied research. This paper examines a specific aspect of stochastic dominance: testing conditional stochastic dominance (CSD) at specific values of the conditioning covariate, referred to as target points. Such conditional comparisons are crucial in many contexts, including evaluating treatment effects in social programs within a regression discontinuity design, analyzing economic disparities across demographic groups, and investigating potential discrimination in decision-making processes.
Unconditional stochastic dominance methods, which analyze entire distributions, have been extensively studied and widely applied in the literature, with foundational contributions dating back to hodges:1958 and mcfadden:1989 and more recent developments in abadie:2002, barrett/donald:2003, Linton/maasoumi/whang:2005, and linton/song/whang:2010, among others. However, in many empirical settings, the primary interest lies in dominance conditional on a subset of the population defined by specific characteristics or values of a conditioning variable. For instance, in regression discontinuity designs, the nature of the methodology necessitates comparing outcome distributions conditional on the cutoff of the running variable (donald/hsu/barrett:2012, shen/zhang:16, goldman/kaplan:2018, qu/yoon:2019). Likewise, in wage discrimination studies, researchers may seek to compare wage distributions across demographic groups while controlling for observed skill levels (becker:1957, canay/etal:2024, bharadwaj/deb/renou:24).
The primary goal of this paper is to test whether the conditional cumulative distribution function (CDF) of one variable stochastically dominates that of the other at specific values of a conditioning variable. Formally, we consider the null hypothesis \[ H_0: F_Y(t | z) \leq F_X(t | z) \quad \text{for all } (t, z) \in \mathbf{R} \times \mathcal{Z}~, \] against the alternative that there exists some \((t, z)\) for which the reversed inequality holds strictly. Here, \( F_Y(\cdot | z) \) and \( F_X(\cdot | z) \) represent the conditional CDFs of the random variables \( Y \) and \( X \), respectively, given \( Z = z \). Importantly, we focus on situations where the set of target values, $\mathcal{Z}$, is not the entire support of $Z$, but rather a finite collection of points (including the case of a singleton). To address this testing problem, we propose a novel procedure that leverages induced order statistics based on independent samples from \( Y \) and \( X \). Our test statistic, a Kolmogorov--Smirnov-type measure, captures the maximal deviation between the empirical CDFs of the two samples, conditional on observations near the target points. Crucially, the critical value we propose is derived in a deterministic, non-data-dependent manner once a tuning parameter is accounted for, ensuring computational simplicity.
Our contributions in this paper are both methodological and theoretical. First, we introduce a novel test for CSD at target points, particularly suited for settings where researchers seek to compare distributions conditional on covariates at specific values of the conditioning covariates. The proposed test exploits induced order statistics, leveraging observations closest to the target conditioning point to construct empirical CDFs that form the basis of our test statistic. Unlike traditional methods, our approach neither relies on kernel smoothing nor imposes parametric assumptions on conditional distributions. However, it requires a tuning parameter, which serves a role analogous to bandwidth selection in nonparametric estimation.
Second, we establish the asymptotic validity of our test under two alternative asymptotic frameworks. In the first framework, the numbers of effective “local” observations, $q_y$ and $q_x$, are held fixed as the sample size $n \to \infty$. In this setting, we show that the test statistic converges to a limiting experiment in which the induced order statistics behave as independent draws from the true conditional CDFs at the target point. This convergence yields a feasible critical value that remains valid without relying on resampling methods such as the bootstrap. Our regularity conditions in this fixed-$q$ framework are mild: the conditional CDFs at the target point may contain finitely many discontinuities; both $Y$ and $X$ may be continuous, discrete, or mixed; and we require only that the conditional CDFs $F_Y(t \mid z)$ and $F_X(t \mid z)$ be continuous in $z$ at the target point, uniformly in $t$. This is a weaker condition than the smoothness assumptions, such as twice differentiability in $z$, typically imposed in nonparametric methods. In the second framework, the numbers of effective observations $q_y$ and $q_x$ are allowed to diverge as $n \to \infty$. Building on recent results on convergence rates for induced order statistics bugni/canay/kim:26b, we derive an explicit rate at which the rejection probability of our test approaches its nominal level as a function of $q_y$, $q_x$, and $n$. This second framework complements the fixed-$q$ results by demonstrating that the test continues to control size when the number of effective observations increases, and it provides practical guidance for the data-dependent selection of the tuning parameters $q_y$ and $q_x$.
Third, we show that the proposed critical value aligns with the one obtained from a permutation-based approach when the random variables $Y$ and $X$ are both continuous, thus establishing a natural connection between our method and the broader literature on permutation-based inference. To the best of our knowledge, this result provides the first formal justification for the validity of permutation-based inference in testing unconditional stochastic dominance. We demonstrate that the critical value of our test cannot be improved when both $Y$ and $X$ are continuous. However, this result does not extend to the case when either $Y$ or $X$ is discretely distributed. For this latter case, we introduce a refined critical value, which is typically smaller than the default one we propose and only a function of the support points for $Y$ and $X$. This refinement enhances power relative to the default critical value, though it comes at the cost of increased computational complexity.
Finally, we examine the finite-sample performance of our test through Monte Carlo simulations and present an empirical application to illustrate its implementation. Both exercises suggest that the data-dependent rule we propose for selecting the key tuning parameters performs well in practice.
Our work contributes to the extensive literature on stochastic dominance testing, building on seminal contributions such as anderson:1996, davidson/duclos:2000, abadie:2002, barrett/donald:2003, Linton/maasoumi/whang:2005, and linton/song/whang:2010. These studies examine the null hypothesis of unconditional stochastic dominance and predominantly rely on asymptotic arguments and resampling techniques. Our approach to testing CSD differs in that, in the limit experiment, the conditional testing problem simplifies to a finite-sample unconditional testing problem. In the context of CSD testing, prior research---including delgado/escaciano:2013, gonzalo/olmo:2014, chang-lee-whang-2015-EJ, and andrews-shi-2017-JOE---evaluate stochastic dominance over a range or the entire support of a continuous conditioning variable. Our work diverges from this literature by testing CSD at specific target points. A distinct line of research, including donald/hsu/barrett:2012, shen/zhang:16, goldman/kaplan:2018, and qu/yoon:2019, studies CSD within regression discontinuity designs, where dominance is defined conditional on cutoffs. All of these methods assume continuity of conditional distributions, limiting their applicability in empirical settings featuring discrete mass points. Our method relaxes this constraint and accommodates distributions with finitely many discontinuities. The resulting test is novel, computationally simple, and valid across a broader class of distributions.
Our work closely aligns with the well-established literature on testing the equality of two continuous distributions. Foundational contributions by gnedenko/korolyuk:1951, korolyuk:1955, and blackman:1956 established that the finite-sample distribution of the two-sample one-sided Kolmogorov--Smirnov test statistic is pivotal under the null, deriving closed-form expressions under various simplifying assumptions. Later research by hodges:1958, hajek/sidak:1967, and durbin:1973 developed methods to approximate this finite-sample distribution. Although our null hypothesis differs, our critical value coincides with the corresponding quantile of this distribution. Notably, hodges:1958 and mcfadden:1989 proposed that the two-sample one-sided Kolmogorov--Smirnov test, originally designed for testing equality of two distributions, could be adapted to test stochastic dominance under continuity assumptions, though without formal proof. We provide a rigorous justification for this claim.
The remainder of the paper is organized as follows. Section (ref) defines the testing problem and introduces notation. Section (ref) presents the induced order statistics that form the basis of our procedure and introduces the proposed test. Section (ref) establishes its asymptotic validity under two alternative asymptotic frameworks. Section (ref) discusses several refinements and extensions, including results on rates of convergence, a data-dependent rule for selecting the tuning parameters, and a refinement of the test that increases power when $Y$ and $X$ are discrete. Importantly, Section (ref) also provides formal results on the validity of permutation tests for stochastic dominance with continuously distributed random variables, offering novel arguments for the validity of permutation-based inference in this setting. Section (ref) evaluates the finite-sample performance of our test through Monte Carlo simulations. Finally, Section (ref) concludes. All proofs are contained in the Appendix.
Let $(Y,Z)$ and $(X,Z)$ denote random pairs, each supported on $\mathbf{R}\times\mathbf{R}$. Let $P_Y$ and $P_X$ denote the joint distributions of $(Y,Z)$ and $(X,Z)$, respectively. For $t\in\mathbf{R}$ and $z$ in the support of $Z$, define the conditional distribution functions \[ F_Y(t | z) := P_Y\{Y \le t \mid Z=z\} \qquad\text{and}\qquad F_X(t | z) := P_X\{X \le t \mid Z=z\}~. \]
We are interested in testing the null hypothesis:
versus the alternative hypothesis:
where $\mathcal{Z} = \{z_1, \dots, z_L\}$ is a finite set of target values of the conditioning variable. The case where $L = 1$ and $Z$ is a continuously distributed random variable is both simpler and particularly relevant in empirical applications. To minimize notational clutter, we focus on this case for most of the remainder of the paper, with Section (ref) addressing the case where $L > 1$.
To test this hypothesis, we assume the analyst observes two independent samples:
We refer to the first sample as the $Y$-sample, which consists of $n_y$ i.i.d.\ draws from $P_Y$, and to the second sample as the $X$-sample, which consists of $n_x$ i.i.d.\ draws from $P_X$. Independence of the samples implies that the data-generating distribution is the product measure \[ P := P_Y \otimes P_X~. \] All probability and expectation statements in what follows are taken with respect to $P$ unless otherwise noted. We denote by $\mathbf P$ the class of data-generating distributions $P$ satisfying the regularity conditions imposed later, and let
denote the subset of distributions $P\in \mathbf P$ satisfying the null hypothesis in (ref).
Let the observed data be those given in (ref). Let $(q_y,q_x)$ be two small positive integers (relative to $(n_y,n_x)$) and consider the point $z_{0}\in \mathcal Z$. The test we propose is based on the following two samples:
To define these samples formally, we introduce $g$-order statistics for the conditioning variable $Z$, where $g(Z) := |Z - z_{0}|$; see reiss:89 and kaufmann/reiss:92. For any two values \( z, z' \in \mathcal{Z} \), we define the ordering \(\le_{g}\) as follows: \[ z \le_{g} z' \quad \text{if and only if} \quad g(z) \le g(z')~. \] This defines a $g$-ordering on the set $\mathcal{Z}$. The $g$-order statistics \( Z_{g,(i)} \) are then the values of \( Z \) ordered according to this criterion: \[ Z_{g,(1)} \le_{g} Z_{g,(2)} \le_{g} \cdots \le_{g} Z_{g,(n)}~. \] If there are ties in the $g$-ordering, they can be resolved arbitrarily—for example, by relying on the original sample order.
We then take the values of $\{Y_i:1\le i\le n_y\}$ associated with the $q_y$ smallest $g$-ordered statistics of $Z$ in the $Y$-sample, denoted by
That is, $Y_{n_y,[j]}=Y_k$ if $Z_{g,(j)}=Z_k$ for $k=1,\dots,q_y$. Similarly, we take the values of $\{X_i:1\le i\le n_x\}$ associated with the $q_y$ smallest $g$-ordered statistics of $Z$ in the $X$-sample, denoted by
The random variables in (ref) and (ref) are referred to as induced order statistics or concomitants of order statistics; see david/galambos:74,bhattacharya:74,canay/kamat:18. Intuitively, we view these samples as independent samples of \(Y\) and \(X\), conditional on \(Z\) being `close' to \(z_{0}\). A key feature of our test is that it relies solely on these induced order statistics. Letting \(n := \min\{n_y,n_x\}\) and \(q := q_y + q_x\), the effective (pooled) sample is
It is also important to note that the first \(q_y\) elements of \(S_n\) are associated with the \(Y\)-sample, while the remaining \(q_x\) elements come from the \(X\)-sample.
Having defined the induced order statistics, we now define our test statistic as
where the empirical CDFs are
The test statistic in (ref) is a two-sample one-sided Kolmogorov--Smirnov (KS) test statistic, see hajek/sidak/sen:99, and the test we propose rejects the null hypothesis in (ref) when $T(S_n)$ exceeds a critical value, defined next.
To introduce the critical value, let $\alpha\in(0,1)$ be given and $\{U_j: 1\le j\le q\}$ be a sequence of uniform random variables i.i.d., that is, $U_j\sim U[0,1]$. Define $\Delta(u)$ as
and
The test we propose for the null hypothesis in (ref) is
We reiterate that (ref) corresponds to our test with a single target point ($L = 1$). Section (ref) extends this framework to the general case with multiple target points ($L > 1$). Furthermore, in addition to our default critical value in (ref), we provide a refined critical value tailored for discrete $Y$ and $X$ with finite support in Section (ref).
The choice of the two-sample one-sided KS test statistic in (ref) is crucial for accommodating cases where \( Y \) and \( X \) are discrete or mixed. While one could construct analogues of our test in (ref) using alternative statistics commonly employed in the stochastic dominance literature---such as one-sided versions of the Cram\'er--von Mises or Anderson--Darling statistics---our asymptotic validity result does not generally extend to these alternatives. See the discussion following Theorem (ref).
In this section, we derive the asymptotic properties of the test in (ref) using two alternative asymptotic frameworks. The first one requires $q := q_y+q_x$ to be fixed as $n\to \infty$ and represents a finite sample situation where the effective number of observations used by the test is too small to credibly invoke approximations that require a “large” value of $q$. The second framework requires $q\to \infty$ slowly as $n\to \infty$, and represents a finite sample situation where the effective number of observations used by the test is large enough to invoke approximations for “large” $q$.
In this section, we examine the asymptotic properties of the test in (ref) within a framework where $q := q_y+q_x$ is fixed and $n:=\min\{n_y,n_x\}\to \infty$. We first derive the asymptotic properties of induced order statistics in (ref), and then present our main theorem.
We start by deriving a result on the induced order statistics collected in the vector $S_n$ in (ref). To do so, we make the following assumptions.
Assumption (ref) requires that the distribution of $Z$ is locally dense at each of the points in $\mathcal{Z}$. This includes the case where $Z$ has a mass point at $z\in \mathcal Z$. Assumption (ref) is a smoothness assumption required to guarantee that conditioning on observations close to $z$ is informative about the distribution conditional on $Z=z$.
Theorem (ref) is a special case of Theorem (ref) in the appendix with $L=1$, which generalizes canay/kamat:18 to accommodate multiple conditioning values. It establishes that the limiting distribution of the induced order statistics in the vector $S_n$ is such that the elements of the vector, denoted by $S$, are mutually independent. Specifically, the first $q_y$ elements of this vector follow the distribution $F_Y(\cdot|z_{0})$, while the remaining $q_x$ elements follow $F_X(\cdot|z_{0})$. The proof leverages the fact that the induced order statistics $S_n$ in (ref) are conditionally independent given $(Z_1,\dots,Z_n)$, with conditional CDFs $$F_Y(\cdot|{Z_{n_y,(1)}}),\dots,F_Y(\cdot|{Z_{n_y,(q_y)}}),F_X(\cdot|{Z_{n_x,(1)}}),\dots,F_X(\cdot|{Z_{n_x,(q_x)}})~.$$ The result then follows by showing that $Z_{n_y,(j)}\overset{p}{\to}z_0$ and $Z_{n_x,(j)}\overset{p}{\to}z_0$ for all $j\in\{1,\dots,q\}$, and invoking standard properties of weak convergence. Theorem (ref) plays a crucial role in our asymptotic validity result presented in Theorem (ref).
In addition to Assumptions (ref) and (ref), we also require that, conditional on $Z=z$, the random variables $Y$ and $X$ have distributions with a finite number of discontinuity points. To state this assumption formally, let $\mathcal{D}_{Y}(z)$ and $\mathcal{D}_{X}(z)$ denote the sets of discontinuity points of the CDFs of ${ Y|Z=z }$ and ${ X|Z=z }$, respectively.
It is important to note that Assumption (ref) allows both $Y$ and $X$ to be continuous, discrete, or mixed random variables. However, it excludes cases where these variables have countably many discontinuities conditional on $z\in \mathcal Z$. We also point out that Theorem (ref) does not require Assumption (ref).
We now formalize our main result in Theorem (ref), which shows that the test defined in (ref) is asymptotically level $\alpha$ under the assumptions we just introduced. Below, we denote by $E_P[\cdot]$ the expected value with respect to the distribution $P\in \mathbf P$.
Theorem (ref) establishes the asymptotic validity of the test in (ref). There are three main reasons why the inequality in (ref) may be strict, resulting in the limiting rejection probability strictly below $\alpha$. First, for distributions $P$ in the interior of $\mathbf{P}_0$, where the inequality in (ref) holds strictly for some $t \in \mathbf{R}$, the test is expected to reject with probability less than $\alpha$, with a magnitude depending on the `distance' of $P$ from the boundary of $\mathbf{P}_0$. Second, in cases where $Y$ or $X$ are not continuously distributed, the critical value defined in (ref) serves as an upper bound for the desired quantile, as discussed further in Section (ref). Finally, the test statistic $\Delta(u)$ in (ref) is discretely distributed, taking only a limited number of distinct values. Consequently, the achieved significance level
satisfies $\bar{\alpha} \leq \alpha$ by definition, but may be strictly less than $\alpha$. Whether $\bar{\alpha} = \alpha$ occurs or not depends on whether the critical value $c_\alpha(q_y, q)$ aligns exactly with one of the discrete jumps in the CDF of $\sup_{u \in (0,1)} \Delta(u)$, which depends on $\alpha$, $q_y$, and $q$.
We derive Theorem (ref) by linking the weak convergence of the induced order statistics \( S_n \) to the limit variable \( S \) in (ref) with the finite-sample validity of the test \( \phi(S) \) in the limit experiment. This connection becomes nontrivial when the data are not continuously distributed. The KS statistic plays a central role in addressing these challenges. First, our proof leverages the fact that the rank of the induced order statistics is preserved as the sample size grows. The KS statistic, being rank-based, ensures that the rejection rate of \( \phi(S_n) \) converges to that of \( \phi(S) \) in the limit experiment. Second, the structure of the KS statistic implies that our test controls size in the limit experiment over our class of null distributions---including those that are discrete or mixed---thereby establishing asymptotic validity. By contrast, as shown in Section (ref) in the appendix, analogous results generally fail when using the one-sided versions of the Cramér--von Mises or Anderson--Darling statistics, which are commonly used in the stochastic dominance testing. Nonetheless, in the special case where the data are continuously distributed, we show in Theorem (ref) that the Cramér--von Mises-based test remains valid.
In this section, we examine the asymptotic properties of the test in (ref) within a framework where $q:= q_y+q_x \to \infty $ as $n\to \infty$. Our results here follow from the recently derived rates of convergence for induced order statistics in bugni/canay/kim:26b, and so we keep the discussion brief.
The result in Theorem (ref) establishes that the limiting distribution of the induced order statistics in the vector $S_n$ is such that the elements of the limit vector, denoted by $S$, are mutually independent. Thus, $S_n$ serves as an “approximation’’ to $S$, and while the infeasible test $\phi(S)$ would be ideal, in practice we must work with $\phi(S_n)$. Since $\phi \le 1$, the variational characterization of total variation implies
where $\mathcal L(\cdot)$ denotes the distribution of a random vector and $\mathrm{TV}$ denotes total variation distance. Hence, a bound on the rate of convergence of $\mathrm{TV}\big(\mathcal L(S_n),\mathcal L(S)\big)$ would immediately yield a rate at which the rejection probability $E_P[\phi(S_n)]$ is asymptotically bounded by $\alpha$, since in our setting $ E_P[\phi(S)]\le \alpha$. Unfortunately, Theorem (ref) is silent about the rate for the convergence of $S_n$ to $S$.
Theorem (ref) in Appendix (ref) leverages the results in bugni/canay/kim:26b to show that, under assumptions slightly stronger than those stated in Section (ref), it follows that
and so
Hence, the test defined in (ref) is asymptotically level $\alpha$ under those same assumptions, provided that $q_y = o(n_y^{2/3})$ and $q_x = o(n_x^{2/3})$. We rely on these rates of convergence in the next section to derive data-dependent rules of thumb for selecting the two tuning parameters.
We now discuss the practical considerations for implementing our test. We propose a data-dependent method for two tuning parameters $q_y$ and $q_x$, drawing on arguments from armstrong/kolesar:18, similar to the approach used by bugni/canay:21. Importantly, the analysis from these arguments is consistent with the rates of convergence implied by (ref). Concretely, this method leverages a bias-variance trade-off inherent in the estimation of the conditional CDFs used in the test statistic for $\phi(S_n)$, within an asymptotic framework where $q$ can grow slowly with $n$. Our goal is to provide practical guidance for choosing these tuning parameters based on the data, rather than claiming optimality or even validity of any sort. We examine the performance of this rule via Monte Carlo simulations in Section (ref) and use it in the empirical application in Section (ref).
We propose choosing $q_x$ and $q_y$ using the following data-dependent rules:
and
where $\mu_Z := E[Z]$, $\sigma_Z^2 := \text{Var}[Z]$, $\phi_{\mu_Z,\sigma_Z}(\cdot)$ denotes the probability density function of a normal distribution with mean $\mu_Z$ and variance $\sigma_Z^2$, and $\rho_Y$ and $\rho_X$ are the correlation coefficients between $Y$ and $Z$, and $X$ and $Z$, respectively.
To provide some intuition as to why this rule of thumb may be reasonable, assume that the random variable $Z$ is continuous with a density function $f_Z(\cdot)$ satisfying
and any values $z_1,z_2\in \mathcal Z$. In addition, suppose that the conditional CDF of $Y$ satisfies
and that the conditional CDF of $X$ satisfies the same condition with a constant $C_X$. It can be shown that the standardized bias $B_{n_y,q_y}$ associated with the estimator of the conditional CDF $F_Y(\cdot|z_{0})$ satisfies
Let $t^{\ast}$ denote the right-hand side of (ref). Solving for $q_y$, we obtain
The proposed data-dependent rule in (ref) and (ref) can be viewed as under-smoothed approximations of these values, where the unknown Lipschitz constants are approximated by the working model $Z \sim N(\mu_Z, \sigma_Z)$. This guarantees that $q_y = o(n_y^{2/3})$, which is the condition discussed in Section (ref) to obtain (ref). The constant multiplying $n_y^{1/2}$ in (ref) is intuitive for two reasons. First, it reflects that a steeper density at $z_{0}$, or a steeper derivative of the conditional CDFs at $z_{0}$ calls for smaller $q_y$. In such cases, nearby observations provide a poorer approximation of the quantities at $z_{0}$. Since the maximum slope is determined by the constants $C_Z$ and $C_Y$, the rule is inversely proportional to these constants. Second, the rule accounts for low density at $z_{0}$. When $f_Z(z_{0})$ is small, the $q_y$ nearest observations are likely to be farther from $z_{0}$, again requiring smaller $q_y$. While one could replace the normality assumption with a nonparametric estimator of $f_Z(\cdot)$, it is unfortunately impossible to adaptively choose $C_Z$ and $C_Y$ for testing (ref) armstrong/kolesar:18. Since any data-dependent rule for $q$ must reference $C_Z$ and $C_Y$, we prioritize simplicity and use normality for both $f_Z(\cdot)$ and the associated constants.
While our default critical value in (ref) is valid as long as the distributions of random variables $Y$ and $X$ have finitely many discontinuities, it is possible to construct a smaller refined critical value when both variables are discretely distributed with a limited number of support points. Our test with the refined critical value still maintains the asymptotic validity, though it comes at the cost of additional computational complexity.
To motivate the refined critical value, let $\mathbf Y$ and $\mathbf X$ denote the support of $Y$ and $X$, respectively. Our proof for the asymptotic validity (specifically, Theorem (ref) in the appendix) relies on the following inequalities:
where $\mathcal{U} := \cup_{t\in \mathbf Y}\{u = F_X(t|z_{0}) \}$ is the set of values that $F_X(t|z_{0})$ takes as $t$ varies over $\mathbf Y$ and $\Delta(u) $ is given in (ref). Once we replace the set $ \mathcal{U}$ with the interval $(0,1)$, our default critical value $c_{\alpha}(q_y,q)$, defined as a quantile of $\sup_{u\in(0,1)}\Delta(u)$ in (ref), ensures the probability is bounded below $\alpha.$ However, replacing the set $ \mathcal{U}$ with $(0,1)$ could be unnecessarily conservative when $ \mathcal{U}$ contains only a few points---either because $\mathbf Y$ contains only a few points or because $F_X(t|z_{0})$ takes few distinct values as $t$ varies. In fact, the cardinality of the set $ \mathcal{U}$ is determined by the smaller support size of $Y$ and $X$.
To define our refined critical value, let $r$ denote the smaller support size of $Y$ and $X$,
and let $\mathbf{U}_r$ denote the collection of all ordered $r$-tuples of distinct points in (0,1),
We denote an arbitrary element of $\mathbf{U}_r$ by $\mathcal{U}_r$. Our refined critical value is defined as
with $\Delta(u)$ as in (ref), and the refined test for the null hypothesis in (ref) when either $Y$ or $X$ is discretely distributed with a limited number of support points is thus
Section (ref) presents the general version of this test for the case $L>1$.
The power gains of using $c_{\alpha}^{r}(q_y, q)$ over $c_{\alpha}(q_y, q)$ are most pronounced when $r$ is small. Our numerical analysis shows the largest gains for $r \le 10$. Thus, this refinement is most effective for discrete data with a limited number of support points, rather than all discrete settings. Moreover, the computational cost of $c_{\alpha}^{r}(q_y, q)$ increases with $r$, and so as $r$ grows, the gains diminish while the cost rises.
We propose to compute $c_{\alpha}^{r}(q_y, q)$ numerically, by solving the following optimization problem:
where $\mathbf{T}$ is the support of $\Delta(u)$ in (ref). Here, $c_{\rm ub} = c_{\alpha}(q_y, q)$ and $c_{\rm lb}$ is given by
The fact that $c_{\alpha}(q_y, q)$ provides a valid upper bound is unsurprising given the preceding discussion. On the other hand, $c_{\rm lb}$ serves as a valid lower bound because $\left\{\frac{1}{1+r},\frac{2}{1+r},\dots,\frac{r}{1+r} \right\}$ is a specific element in $\mathbf U_{r}$. In our numerical evaluations, we often found that $c_{\alpha}^{r}(q_y, q) = c_{\rm lb}$, but not always. This indicates that the additional optimization in (ref) cannot be generally avoided. For modest values of $r$ and $q$, however, this optimization step is computationally straightforward, primarily due to the relatively small number of points typically found in $[c_{\rm lb}, c_{\rm ub}] \cap \mathbf{T}$.
In this section, we study the properties of the test $\phi(\cdot)$ in (ref) in the limit experiment associated with the asymptotic framework in Section (ref). By Theorem (ref), this test is equivalent to $\phi(S)$, where
In words, in the limit experiment, we observe one random sample of size $q_y$ from the distribution $F_Y(\cdot|z_{0})$ and the other independent random sample of size $q_x$ from the distribution $F_X(\cdot|z_{0})$. The KS test statistic in (ref) is a function of $S$, and the critical value $c_{\alpha}(q_y, q)$ remains unchanged. We begin our discussion by focusing on the case where $S$ is continuously distributed.
The finite-sample properties of the two-sample one-sided KS statistic have been extensively studied in the literature of testing equality of two continuous (unconditional) distributions. Early works established that the test statistic's finite-sample distribution is pivotal under the null and developed algorithms for its computation (gnedenko/korolyuk:1951,korolyuk:1955,blackman:1956,hodges:1958,hajek/sidak:1967, and durbin:1973). Our critical value is obtained from this pivotal finite-sample distribution, despite the different null hypothesis.
This connection to the literature of testing equality of two distributions arises from the observation that the distribution \(F_Y(\cdot|z_{0}) = F_X(\cdot|z_{0})\) is the least favorable within the set of null distributions \(\mathbf{P}_0\) in (ref) satisfying stochastic dominance, a point first made by lehmann:1951 and later reiterated by hodges:1958,mcfadden:1989, and goldman/kaplan:2018. We define the subset of continuous distributions in \(\mathbf{P}_0\) that satisfy \(F_Y(\cdot|z_{0}) = F_X(\cdot|z_{0})\) as \(\mathbf{P}^*_0\), and denote a generic element in \(\mathbf{P}^*_0\) by \(P^*\). Although these papers studied the distribution of KS statistic under \(P^*\), they did not provide a formal proof that \(P^*\) determines the size of the test under the null hypothesis in (ref). For completeness, we formally state this result in Lemma (ref) below, and provide its proof.
Lemma (ref) shows that when $S \sim P^*\in \mathbf P^*_0$, our test is `exact' in that it achieves the closest possible rejection rate to $\alpha$, defined as $\bar{\alpha}$ in (ref). In this case, we have $$ T(S) \stackrel{d}{=} T(U) \quad \text{where}\quad \{U_j \sim U[0,1]: 1 \leq j \leq q\} \; \text{are i.i.d.}$$ This demonstrates that the analytical (finite-sample) critical value $c_{\alpha}(q_y, q)$ for the one-sided KS test can be accurately approximated by simulating uniform random variables. However, when $S \sim P\in \mathbf P_0$ is such that $P\{S_i = S_{j}: i\not=j\} > 0$, the connection $T(S) \stackrel{d}{=} T(U)$ breaks down. In this case, $c_{\alpha}(q_y, q)$ is no longer the finite-sample analytical quantile of $T(S)$, but rather a valid upper bound.
When \(S\) is continuously distributed, the proposed test \(\phi(S)\) is equivalent to a non-randomized permutation test. This connection establishes the validity of permutation tests for testing stochastic dominance. While hodges:1958 and mcfadden:1989 suggested that a permutation test could be used for this purpose, they did not provide a formal justification. To the best of our knowledge, our proof of this result is novel.
To formally define a permutation test, we introduce the following notation. Let \(\mathbf{G}\) denote the set of all permutations \(\pi = (\pi(1), \dots, \pi(q))\) of \(\{1, \dots, q\}\). The permuted values of \(S\) are given by \[ S^{\pi} = (S_{\pi(1)}, \dots, S_{\pi(q)})~. \] The (non-randomized) permutation test is then defined as follows:
It follows from standard arguments (see, e.g., lehmann/romano:05) that when $S$ is invariant to permutations, i.e., $S \stackrel{d}{=} S^{\pi}$, the randomized version of the test $\phi^{\rm p}(S)$ is exact in finite samples. However, under the null hypothesis in (ref), we have that $S \stackrel{d}{\not =} S^{\pi}$ for some $P \in \mathbf{P}_0$, and so invariance (or the so-called randomization hypothesis) fails. Therefore, the traditional finite-sample arguments for validity no longer apply. Alternative arguments that claim validity of permutation tests when invariance does not hold typically require $q \to \infty$; see chung/romano:13,canay/romano/shaikh:17, and bugni/canay/shaikh:18, among others. In our current setting, where $q$ is fixed, such arguments do not apply.
We contribute to this literature by demonstrating that, when \(S\) is continuously distributed, our test is equivalent to a non-randomized permutation test. The formal statement of this result follows.
Lemma (ref) shows that our data-independent critical value in (ref) is equivalent to the critical value of a permutation test when the random variable $\tilde{S}$ has no ties. Lemma (ref) immediately implies when $S_n$ in (ref) is continuously distributed, we have
It follows from (ref) and Theorem (ref) that, when $S$ is continuously distributed, a non-randomized permutation test controls the limiting rejection probability under the null hypothesis in (ref). Importantly, this result holds even though invariance does not hold for all $P \in \mathbf{P}_0$.
The results in Lemmas (ref) and (ref) reveal interesting and novel connections between our test and classical arguments involving finite-sample critical values and permutation tests. However, these results critically depend on the random variable $S$ being continuously distributed and do not extend to cases where $S$ is discretely distributed.
When \( S \) is discrete and ties occur with positive probability, i.e., \( P\{S_i = S_{j}\} > 0: i\not=j \), the finite-sample distribution of the KS test statistic \( T(S) \) depends on the number and location of these ties. A natural approach to handle ties is to redefine the test statistic to randomly break them, effectively making the test a randomized one. While this would allow us to establish an analog of Lemma (ref) for discrete data, we do not pursue such an extension, as randomized tests are rarely used in practice. Despite our best efforts, we were unable to demonstrate that a permutation test could control size under the null hypothesis of stochastic dominance in (ref) when $S$ is discrete, without relying on random tie-breaking rules.
In this section, we evaluate the finite-sample performance of the test in (ref) for $L=1$ or the test proposed in Section (ref) for $L>1$ through a simulation study. We present a variety of data-generating processes to illustrate both the strengths and potential limitations of our test.
We consider seven distinct designs with four cases, (a) to (d), in each design as follows:
The first three designs are based on the following location–scale model:
where $U$ and $V$ are random variables with specified distributions, and the conditioning variable $Z$ follows a non-negative Beta$(2,2)$ distribution. Under this location–scale model, whenever $\sigma_Y(z_{\ell}) = \sigma_X(z_{\ell})$, the null hypothesis in (ref) holds as long as
Design 1 satisfies (ref) for all $z\in(0,1)$. Design 2 satisfies (ref) at $z_{\ell} = 0.5$, but violates the inequality for $z > 0.5$, which may affect the performance of our test in finite samples due to its reliance on induced order statistics. Design 3 is such that $U$ and $V$ are $U[0,1]$, which guarantees that (ref) holds even when $\sigma_Y(z_{\ell}) < \sigma_X(z_{\ell})$, provided $\mu_Y(z_{\ell}) - \mu_X(z_{\ell}) \ge \sigma_X(z_{\ell}) - \sigma_Y(z_{\ell})$.
Design 4 is a slight variation of the location-scale model to accommodate a regression discontinuity design (RDD):
Following shen/zhang:16, we consider the case where $U$ and $V$ are independent $N(0,1)$ random variables, $Z \sim 2\mathrm{Beta}(2,2) - 1$, and both $\mu_Y(Z)$ and $\mu_X(Z)$ are defined in terms of the function
A key feature shared by Designs 1 through 4 is that both $X$ and $Y$ are continuously distributed; a feature not shared by the next three designs.
Design 5 is such that $U$ and $V$ follow log-normal distributions with the bottom 20% censored. This setup reflects features of wage distributions, which often exhibit a point mass at the minimum wage. Design 6 defines conditional probabilities \( P\{X = k | Z\} \) and \( P\{Y = k | Z\} \) as $$ P\{X = k | Z\} = \frac{e^{\theta^x_k(\frac{3}{2}-Z)}}{\sum_{j=1}^3 e^{\theta^x_j(\frac{3}{2}-Z)}} ~~\text{and}~~ P\{Y = k | Z\} = \frac{e^{\theta_k^y(\frac{3}{2}-Z)}}{\sum_{j=1}^3 e^{\theta^y_j(\frac{3}{2}-Z)}} ~\text{for }k=1,2,3, $$ where, for $\mu_Y(z) \in [-1,1]$,
The parameter $ \theta^x_k = (-0.5, -1.5, -2)$ controls the baseline log-odds of each category for $X$, while the factor $(3/2 - Z)$ introduces a monotonic dependence on $Z$. Finally, Design 7 considers $$ X|Z \sim B\Big([25Z], \frac{1}{2}\Big)\quad \text{and}\quad Y|Z \sim B\Big([25Z] + \mu_Y(Z), \frac{1}{2}\Big)~,$$ where $B(\cdot, \cdot)$ denotes a binomial distribution and $[x]$ represents the nearest integer to $x$.
The parameter values for all Designs are reported in Appendix (ref), where we also report the mean values of \(q_y^*\) and \(q_x^*\) across simulations.
We report results for sample size $n = 1,000$ and nominal level $\alpha = 10\%$, based on $10,000$ Monte Carlo simulations to test the null hypothesis in (ref). To implement the test $\phi(\cdot)$ in (ref), denoted by `KS' in the figures, we select the tuning parameters $(q_y, q_x)$ using the data-dependent rules in (ref) and (ref). For Designs 1-3 with $L=1$, we compare our test with the one proposed in goldman/kaplan:2018. For the RDD Design 7 with $L=1$, we compare our test with the method in shen/zhang:16.\footnote{SZ is implemented using the authors’ rule of thumb, whereas for GK we use our own rule of thumb, as the original paper does not provide guidance on how to select their tuning parameter. Note that neither of these tests is defined when $L>1$.} For the mixed and discrete designs, we present results for our test in (ref) as well as for its refined version in (ref).
Figure (ref) reports rejection probabilities under the null hypothesis for the continuously distributed designs. When the data-generating process satisfies the null in (ref) with equality (case (a)), the KS test’s rejection probabilities closely align with the nominal level and consistently outperform the GK and SZ tests. When the null holds with strict inequality (case (b)), rejection rates fall below \(\alpha\), consistent with our critical value serving as a valid upper bound for the true quantile. Case (c), where \(L = 2\), shows behavior similar to case (a), where \(L = 1\). Overall, the KS test exhibits excellent size control.
Figure (ref) reports rejection probabilities under the null hypothesis for the mixed and discretely distributed designs. When the data-generating process satisfies the null in (ref) with equality, the KS test we propose in (ref) works well when the data is mixed, but it is conservative when the data is discrete with few support points. Designs 6 and 7 illustrate that the refined critical value \(c^{\rm r}_{\alpha}(q, q_y)\) in (ref) offers a more accurate approximation of the true quantile of the KS test statistic, though it may still be somewhat conservative.
Figure (ref) reports rejection probabilities under the alternative hypothesis for all designs. The power of the KS test is similar to that of GK and could be above, below, or roughly the same. The power of the KS test is much higher than that of SZ for this design. This is worth noting given that the KS test uses about $90$ observations (for both $q_y$ and $q_x$) while the SZ test uses $276$ effective observations on each side of the threshold. The power results also demonstrate that the refined critical value \(c^{\rm r}_{\alpha}(q, q_y)\) in (ref) enhances power in discrete cases.
This paper introduces a novel test for conditional stochastic dominance (CSD) at target points, offering a flexible, nonparametric approach that avoids kernel smoothing while ensuring computational efficiency. By leveraging induced order statistics, our method constructs empirical CDFs using observations closest to the target conditioning point. We establish the asymptotic properties of our test, demonstrating its validity under weak regularity conditions, and derive a critical value that eliminates the need for resampling techniques such as the bootstrap. Additionally, we extend our framework to better handle discrete data, proposing a refined critical value that enhances the power of the test with minimal additional information. Monte Carlo simulations align with our theoretical results and suggest that our test performs well in finite samples, making our test readily applicable to empirical research in economics, finance, and public policy.
An important feature of our test is its simplicity. Once the key tuning parameters are computed, the test only requires a standard test statistic with a deterministic critical value, without the need for kernels, local polynomials, bias correction, or bandwidth selection. Furthermore, our test admits a clear interpretation in the limit experiment, which allows us to connect it with classical analytical critical values and permutation-based tests. In this sense, our findings contribute to the broader literature on stochastic dominance testing by refining conditional inference methods and establishing new links between permutation-based and rank-based approaches. One open question we did not address in this paper concerns the validity of permutation-based tests for the hypothesis of stochastic dominance when both random variables, $Y$ and $X$, are discrete. Despite attempts to formalize this result, we were unable to prove or disprove it. Extensive Monte Carlo simulations (not reported here) suggest that the test may be valid, and this is an area we plan to explore further.
\appendices \setcounter{equation}{0}
By Theorem (ref),
where the elements of $S$ are independent, and $S_{j}\sim F_{Y}(\cdot |z_{0})$ for $j=1,\cdots ,q_{y}$ and $S_{j}\sim F_{X}(\cdot |z_{0})$ for $ j=q_{y}+1,\cdots ,q$. By the almost-sure representation theorem, we have a sequence of random vectors $\{\Tilde{S}_{n}: 1\le i\le \infty \}$ and a random vector $\Tilde{S}$ defined on a common probability space $(\Omega ,\mathcal{A },\Tilde{P})$ such that
Let $R(s)$ denote the rank of $s$, which maps $s$ to a permutation of $\{1,2,\dots ,q\}$. Define the event that the rank of the two vectors coincides as follows,
To reach the conclusion, it suffices to show that
To see this, consider the following argument,
where (1) holds by $\Tilde{S}_{n}\overset{d}{=}S_{n}$, (2) by the fact that $T$ is invariant to rank-preserving transformations, (3) by (ref) and $\phi(\cdot)\in \{0,1\}$, and (4) by $P \in {\bf P}_0$ and Theorem (ref).
We devote the remainder of the proof to establishing (ref). Define \(\mathcal{D} = \mathcal{D}_{X}(z_{0}) \cup \mathcal{D}_{Y}(z_{0})\), where \(\mathcal{D}_{X}(z_{0})\) and \(\mathcal{D}_{Y}(z_{0})\) denote the sets of discontinuity points as specified in Assumption (ref). For any \(\varepsilon > 0\), let
Observe that
To establish this, consider the following argument. For any \(i,j = 1, \dots, q\), there are three possible cases: (i) \(\tilde{S}_{i} < \tilde{S}_{j}\), (ii) \(\tilde{S}_{i} > \tilde{S}_{j}\), or (iii) \(\tilde{S}_{i} = \tilde{S}_{j}\). First, consider case (i), where \(\tilde{S}_{i} < \tilde{S}_{j}\). Under \(E_{1}(\varepsilon)\), this implies \(\tilde{S}_{i} < \tilde{S}_{j} - \varepsilon\). Under \(E_{n,2}(\varepsilon)\), we have \(\tilde{S}_{n,i} - \varepsilon/2 < \tilde{S}_{i}\) and \(\tilde{S}_{j} < \tilde{S}_{n,j} + \varepsilon/2\). Combining these inequalities yields \(\tilde{S}_{n,i} < \tilde{S}_{n,j}\), as required. Case (ii) follows identically by reversing the roles of \(i\) and \(j\). Finally, consider case (iii), where \(\tilde{S}_{i} = \tilde{S}_{j}\). Under \(E_{1}(\varepsilon)\), this implies \(\tilde{S}_{i} = \tilde{S}_{j} \in \mathcal{D}\). By \(E_{n,3}\), it follows that \(\tilde{S}_{n,i} = \tilde{S}_{n,j} \in \mathcal{D}\). Since this argument holds for all \(i, j = 1, \dots, q\), we conclude that \(F_{n}\) follows, as desired.
By (ref), (ref) follows that there exits $\varepsilon >0$ such that
For arbitrary $\delta >0$, (ref) follows from finding $ \varepsilon =\varepsilon (\delta )>0$ and $N(\delta) $ such that $\tilde{P}\{ E_{1}(\varepsilon )\cap E_{n,2}(\varepsilon )\cap E_{n,3}\} \geq 1-\delta $ for all $n\geq N(\delta) $. Let $\varepsilon_{1}=\inf \{ \Vert \tilde{d}-d\Vert /2:d<\tilde{ d}\in \mathcal{D}\} >0$. By Lemma (ref), $\exists \varepsilon _{2}>0$ such that, for $i\not=j=1,\ldots ,q$,
Finally, set $\varepsilon =\min \{ \varepsilon _{1},\varepsilon _{2}\} >0$ for the remainder of the proof. By elementary arguments, it suffices to show that: (i) $\tilde{P}\{ E_{1}(\varepsilon) ^{c}\} \leq \delta /3$, (ii) $\exists N_{2}(\delta) \in \mathbf{N}$ s.t. $ \tilde{P}\{ E_{n,2}(\varepsilon) ^{c}\} \leq \delta /3$ for all $n\geq N_{2}(\delta) $, and (iii)\ $\exists N_{3}(\delta) \in \mathbf{N}$ s.t. $\tilde{P}\{ E_{n,3}(\varepsilon) ^{c}\} \leq \delta /3$ for all $ n\geq N_{3}( \delta ) $. We divide the rest of the proof into three results.
First, we show that $\tilde{P}\{ E_{1}(\varepsilon) ^{c}\} \leq \delta /3$. To this end, pick $ i\not=j=1,\ldots ,q$ arbitrarily. Note that
where (1) holds by $i\not=j=1,\ldots,q$, ${\tilde{S}}_{i}$ and ${ \tilde{S}}_{j}$ being identically distributed, and $\varepsilon =\min \{ \varepsilon _{1},\varepsilon _{2}\} $. From here, we conclude that
as desired. Second, we show that $\exists N_{2}(\delta) \in \mathbf{N}$ such that $\tilde{P}\{ E_{n,2}(\varepsilon) ^{c}\} \leq \delta /3$ for all $n\geq N_{2}(\delta) $. To see this, note that
By $\Tilde{S}_{n}\overset{a.s.}{\to }\Tilde{S}$, $\exists N_{2}(\delta) $ such that the right-hand side is less than $ \delta /3$, as desired. Finally, we show that $\exists N_{3}(\delta) \in \mathbf{N}$ such that $\tilde{P}\{ E_{n,3}(\varepsilon) ^{c}\} \leq \delta /3$ for all $n\geq N_{3}(\delta) $. To see this, note that
By Lemma (ref), $\exists N_{3}(\delta) $ such that the right-hand side is less than $\delta /3$, as desired. This completes the proof of (ref) and the theorem.
Note that
where (1) holds by $P^{\ast }\in \mathbf{P}_{0}$ and (2) by Theorem (ref). To complete the proof, it suffices to show that $E_{P^{\ast }}[\phi (S)]=\bar{\alpha}$. To this end, consider the following argument:
where (1) holds by (ref), (2) holds by pollard:02 and the same arguments used in the proof of Theorem (ref), (3) follows from the continuity of $F$ guaranteeing that
for $ \Delta(u) := \frac{1}{q_y}\sum_{j=1}^{q_y} I\{U_{j} \le u\}-\frac{1}{q_x}\sum_{j=q_y+1}^{q} I\{U_{j} \le u\}$ as defined in (ref), and (4) by definition of $\bar{\alpha}$ in (ref).
Let $Q:=\{Q_{1},\dots ,Q_{q}\}=\{1,2,\dots ,q\}$ and denote by $Q^{\pi }:=\{Q_{\pi (1)},Q_{\pi (2)}\dots ,Q_{\pi (q)}\}$ the permutation $\pi =(\pi (1),\pi (2),...,\pi \left( q\right) )$ of $Q$. Let $R(s)$ denote the rank of $s$, which maps $s$ to a permutation of $Q$. Since the KS statistic $T(\cdot )$ in (ref) is a rank statistic, it follows that for any $s$
where $T^{\ast }$ is a known function; see hajek/sidak/sen:99. That is, the KS test statistic depends on $S$ only through $R(S)$. Define
We divide the rest of the argument into four steps.
Step 1. For any $s\in \mathbf{R}^{q}$ with $s_{i}\not=s_{j} $ for $i\not=j$, and $c_{\alpha }^{\mathrm{p}}(s)$ as in (ref),
To establish this, consider the following derivation,
Here, (1) follows by (ref), (2) follows since $ s_{i}\not=s_{j}$ for $i\not=j$ implies that $R(s^{\pi })=(R(s))^{\pi }$ and $ R(s)=Q^{\bar{\pi}(s)}$ for some $\bar{\pi}\left( s\right) \in \mathbf{G}$, (3) by $\mathbf{G}=\tilde{\mathbf{G}}:=\{\pi \circ \bar{\pi}(s):\pi \in \mathbf{G}\}$, which follows from the fact that $\mathbf{G}$ is a group, and (4) by (ref).
Step 2. $c_{\alpha }^{\mathrm{p}}=c_{\alpha }\left( q_{y},q\right) $, where $c_{\alpha }\left( q_{y},q\right)$ is defined in (ref).
Let $\{U_i : 1\le i\le q\}$ be i.i.d.\ with $U_{i}\sim U\left( 0,1\right) $ and let $\hat{\pi}$ be a uniformly chosen permutation from $ \mathbf{G}$, independent of $U$. Note that $c_{\alpha }\left( q_{y},q\right) $ is the $\left( 1-\alpha \right) $-quantile of $T(U)$ and, by (ref), $ c_{\alpha }^{\mathrm{p}}$ is the $\left( 1-\alpha \right) $-quantile of the CDF $\frac{1}{\left\vert \mathbf{G}\right\vert }\sum_{\pi \in \mathbf{G} }I\left\{ T^{\ast }(Q^{\pi })\leq x\right\} $. The desired result then follows from noting that $\frac{1}{\left\vert \mathbf{G}\right\vert }\sum_{\pi \in \mathbf{G}}I\left\{ T^{\ast }(Q^{\pi })\leq x\right\} $ is the CDF of $T(U)$, as we show next.
Let $E:=\left\{ U_{i}\not=U_{j}\text{ for }i\not=j\right\} $. For any $x\in \mathbf{R}$, our desired result follows from this derivation:
Here, (1) holds by $U\overset{d}{=}U^{\hat{\pi}}$, (2) and (5) by $P\{E\}=1$, (3) by $\hat{\pi}\perp U$ and $\hat{\pi}$ uniformly chosen in $\mathbf{G}$, and (4) by repeating the arguments used to derive (ref).
Step 3. By Step 1, $\{S_{i}\not=S_{j}$ for $ i\not=j\}\subseteq \left\{ c_{\alpha }^{\mathrm{p}}(S)=c_{\alpha }^{\mathrm{p}}\right\} $, and so $ P\left\{ c_{\alpha }^{\mathrm{p}}(S)=c_{\alpha }^{\mathrm{p}}\right\} =1$ holds by our assumption. By Step 2, $c_{\alpha }^{\mathrm{p}}=c_{\alpha }\left( q_{y},q\right) $. The desired result follows from combining these points.
Step 4. To show the last statement, consider $S=\{1 :1\le j \le q\}$. Then, $T(S^{\pi })=T(S)=0$ for all $\pi \in \mathbf{G}$, and so $c_{\alpha }^{\mathrm{p}}(S)=0$. On the other hand, Step 2 implies $c_{\alpha }^{\mathrm{p}}=c_{\alpha }\left( q_{y},q\right) $, which are positive for typical values of $\left( q_{y},q,\alpha \right) $. For example, $q_{y}=1$, $q=2$, and $\alpha =0.1$ yield $c_{\alpha }^{\mathrm{p}}=c_{\alpha }\left( q_{y},q\right) =0.5.$
For any $\varepsilon>0$, we use $o_{\varepsilon}(1)$ to denote an expression that converges to zero as $\varepsilon \to 0$. Analogously, for any $n \in \mathbf{N}$, we use $o_{n}(1)$ to denote an expression that converges to zero as $n \to \infty$.