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Average Density Estimators: Efficiency and Bootstrap Consistency
Keywords: Semiparametric estimation, efficiency, bootstrap consistency.
JEL: C14.
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Peter Phillips is a towering figure in econometrics. Among other things, his pathbreaking work on nonstationary time series (e.g., \citet*{Phillips_1987_ECMA} and \citet*{Phillips-Perron_1988_Bio} in the case of unit root autoregression and \citet*{Phillips-Durlauf_1986_REStud} and \citet*{Phillips-Hansen_1990_REStud} in the case of cointegration) has forcefully demonstrated that estimators can be useful without having limiting distributions that are \textquotedblleft simple\textquotedblright. In this paper, we show that a similar phenomenon occurs in a seemingly very different setting, namely a canonical semiparametric estimation problem in a model with $i.i.d.$ data.
The specific semiparametric estimation problem we consider is the problem of estimating the average density of a continuously distributed random vector (of which we have a random sample of observations). In that setting, a well known apparent shortcoming of simple \textquotedblleft plug-in\textquotedblright \ estimators is that they have biases that are avoidable and potentially non-negligible. In particular, the biases in question prevent the plug-in estimators from achieving semiparametric efficiency under minimal smoothness conditions. In recognition of this, several methods of \textquotedblleft debiasing\textquotedblright\ have been proposed and have been found to be successful insofar as they give rise to estimators that do achieve semiparametric efficiency under minimal smoothness conditions. (The particular examples given in this paper were obtained by applying and combining ideas from \citet*{Hall-Marron_1987_SPL}, \citet*{Bickel-Ritov_1988_Sankhya}, and \citet*{Powell-Stock-Stoker_1989_ECMA}.)
Recognizing that construction of an estimator is often a means to the end of conducting inference, a natural question is whether existing average density estimators permit valid inference to be conducted under minimal smoothness conditions. In this paper, we answer a specific version of the latter question by investigating whether average density estimators achieve bootstrap consistency under minimal smoothness conditions. Looking at estimators through the lens of the bootstrap is of interest for several reasons, most notably because one can answer questions motivated by inference considerations without having to make additional (and potentially arbitrary) assumptions about the behavior of standard errors (i.e., estimators of nuisance parameters). In other words, because bootstrap consistency (or lack thereof) can be interpreted as a property of an estimator, it has the potential to shed new light on the relative merits of competing estimators. In this paper, we show that average density estimation provides an example where this potential is realized.
To be specific, whereas several distinct approaches to debiasing achieve semiparametric efficiency under minimal smoothness conditions, we find that many of the estimators produced by these approaches fail to achieve bootstrap consistency under minimal smoothness conditions. In contrast, in spite of failing to achieve semiparametric efficiency under minimal smoothness conditions, simple plug-in estimators achieve bootstrap consistency under minimal smoothness conditions. In other words, we find that plug-in estimators enjoy certain nontrivial advantages over some of their debiased counterparts.
The paper proceeds as follows. Section (ref) presents the setup and introduces the formal questions we set out to answer. Studying the most prominent average density estimators, Sections (ref) and (ref) are concerned with efficiency and bootstrap consistency, respectively. Alternative estimators are analyzed in Section (ref). Finally, Section (ref) collects proofs of our main results.
Suppose $X_{1},\ldots,X_{n}$ are $i.i.d.$ copies of a continuously distributed random vector $X\in\mathbb{R}^{d}$ with an unknown density $f_{0}.$ Assuming $f_{0}$ is square integrable, a widely studied estimand in this setting is \[ \theta_{0}=\mathbb{E}[f_{0}(X)], \] the average density. Influential work on estimating $\theta_{0}$ includes \citet*{Hall-Marron_1987_SPL}, \citet*{Bickel-Ritov_1988_Sankhya}, and \citet*{Ritov-Bickel_1990_AoS}; see also \citet*{Gine-Nickl_2008_Bernoulli} and the references therein. In econometrics, estimators of $\theta_{0}$ are often viewed as prototypical examples of two-step semiparametric estimators (in the terminology of \citet*{Newey-McFadden_1994_Handbook}) and therefore provide a natural starting point when attempting to shed light on the properties of two-step semiparametric estimators.
In what follows, we shall explore the extent to which certain prominent estimators of $\theta_{0}$ enjoy one (or both) of two desirable properties. The first of these properties is a very conventional one, namely (semiparametric) efficiency. It is well known (e.g., \citet*[Example 9.5.2]{Pfanzagl_1982_Book} and \citet*{Ritov-Bickel_1990_AoS}) that if $f_{0}$ is bounded, then the efficient influence function $L_{0}$ is well-defined and given by \[ L_{0}(x)=2\{f_{0}(x)-\theta_{0}\}. \] Accordingly, an estimator $\hat{\theta}_{n}=\hat{\theta}_{n}(X_{1} ,\ldots,X_{n})$ of $\theta_{0}$ is said to be efficient if it satisfies
Our analysis will proceed under the following condition on the density.
As alluded to earlier, the assumption that $f_{0}$ is bounded serves the purpose of ensuring that \[ \sigma_{0}^{2}=\mathbb{V}[L_{0}(X)], \] the semiparametric variance bound implied by $\left( \ref{Efficiency}\right) ,$ is well-defined and finite. As pointed out by \citet*{Bickel-Ritov_1988_Sankhya} and \citet*{Ritov-Bickel_1990_AoS}, however, some (additional) assumptions are required on the part of $f_{0}$ for semiparametric efficiency to be achievable. For our purposes, it is convenient and turns out to be sufficient to assume that $f_{0}$ is smooth in the sense that it belongs to $B_{2\infty}^{s}(\mathbb{R}^{d}),$ as that assumption will enable us to employ results from \citet*{Gine-Nickl_2008_PTRF} when showing asymptotic negligibility of certain remainder terms. In particular, and as further discussed below, the magnitude \textquotedblleft smoothing\textquotedblright\ bias of the kernel-based estimators under consideration in this paper turns out to depend on $f_{0}$ through the smoothness of the function $f_{0}^{\Delta}$ given by \[ f_{0}^{\Delta}(x)=\int_{\mathbb{R}^{d}}f_{0}(u)f_{0}(x+u)du. \] Condition D is convenient because it follows from \citet*[Lemma 12] {Gine-Nickl_2008_PTRF} that $f_{0}^{\Delta}$ belongs to the H\"{o}lder space $\mathbf{C}^{2s}(\mathbb{R}^{d})$ whenever $f_{0}$ is bounded and belongs to $B_{2\infty}^{s}(\mathbb{R}^{d})$ with $2s\notin\mathbb{N}.$
The second property of interest is (nonparametric) bootstrap consistency. In the setting of this paper, the most attractive definition of that property is the following. Letting $X_{1,n}^{\ast},\ldots,X_{n,n}^{\ast}$ denote a random sample from the empirical distribution of $X_{1},\ldots,X_{n}$ and letting $\hat{\theta}_{n}^{\ast}=\hat{\theta}_{n}(X_{1,n}^{\ast},\ldots,X_{n,n}^{\ast })$ denote the natural bootstrap analog of $\hat{\theta}_{n},$ the bootstrap is said to be consistent if
where $\mathbb{P}_{n}^{\ast}$ denotes a probability computed under the bootstrap distribution conditional on the data.
To motivate interest in $\left( \ref{Bootstrap consistency}\right) ,$ recall that the (nominal)\ level $1-\alpha$ bootstrap confidence interval for $\theta_{0}$ based on the \textquotedblleft percentile method\textquotedblright\ (in the terminology of \citet*{vanderVaart_1998_Book}) is given by \[ \mathsf{CI}_{n,1-\alpha}^{\mathtt{P}}=\left[ \hat{\theta}_{n}-q_{n,1-\alpha /2}^{\ast}~,~\hat{\theta}_{n}-q_{n,\alpha/2}^{\ast}\right] ,\qquad q_{n,a}^{\ast}=\inf\{q\in\mathbb{R}:\mathbb{P}_{n}^{\ast}[\hat{\theta} _{n}^{\ast}-\hat{\theta}_{n}\leq q]\geq a\}. \] This interval is said to be consistent if
and to be efficient if its end points satisfy
where $\Phi(\cdot)$ is the standard normal cdf. In addition to being \textquotedblleft heuristically necessary\textquotedblright, the bootstrap consistency property $\left( \ref{Bootstrap consistency}\right) $ turns out to be sufficient for $\left( \ref{Consistency of percentile interval}\right) $ and $\left( \ref{Efficiency of percentile interval}\right) $ in the cases of interest in this paper. In turn, the property $\left( \ref{Efficiency of percentile interval}\right) $ implies (by the duality between hypothesis tests and confidence intervals) that efficient two-sided tests of simple hypotheses about $\theta_{0}$ can be based on $\mathsf{CI} _{n,1-\alpha}^{\mathtt{P}}$ whenever the interval is efficient. In other words, the property $\left( \ref{Bootstrap consistency}\right) $ has strong and obvious implications for inference and although those implications may seem more important than bootstrap consistency per se, much of our subsequent discussion of the bootstrap focuses on $\left( \ref{Bootstrap consistency} \right) $ for specificity and because that property seems more \textquotedblleft fundamental\textquotedblright\ than $\left( \ref{Consistency of percentile interval}\right) $ and $\left( \ref{Efficiency of percentile interval}\right) $ in the sense that it is not directly associated with a particular inference method.
At any rate, because the properties of $\hat{\theta}_{n}^{\ast}$ and $\mathsf{CI}_{n,1-\alpha}^{\mathtt{P}}$ are governed solely by (the density $f_{0}$ and) the functional form of $\hat{\theta}_{n},$ the properties $\left( \ref{Bootstrap consistency}\right) ,$ $\left( \ref{Consistency of percentile interval}\right) ,$ and $\left( \ref{Efficiency of percentile interval}\right) $ can all be interpreted as properties of the estimator $\hat{\theta}_{n}$ and one of the main purposes of this paper is to explore the relationship between those properties and the more familiar (efficiency) property $\left( \ref{Efficiency}\right) .$
The (nominal)\ level $1-\alpha$ bootstrap confidence interval for $\theta_{0}$ based on \textquotedblleft Efron's percentile method\textquotedblright\ (in the terminology of \citet*{vanderVaart_1998_Book}) is given by \[ \mathsf{CI}_{n,1-\alpha}^{\mathtt{E}}=\left[ \hat{\theta}_{n}+q_{n,\alpha /2}^{\ast}~,~\hat{\theta}_{n}+q_{n,1-\alpha/2}^{\ast}\right] . \] Suppose $\left( \ref{Bootstrap consistency}\right) $ holds. Then $\mathsf{CI}_{n,1-\alpha}^{\mathtt{E}}$ is consistent if also $\left( \ref{Efficiency}\right) $ holds. On the other hand, and in contrast to $\mathsf{CI}_{n,1-\alpha}^{\mathtt{P}},$ it turns out that in the cases of interest in this paper the interval $\mathsf{CI}_{n,1-\alpha}^{\mathtt{E}}$ is inconsistent when $\left( \ref{Efficiency}\right) $ fails. Partly for this reason, we focus on intervals based on the percentile method.
Suppose $\left( \ref{Efficiency}\right) $ holds. Letting $\hat{\sigma} _{n}^{2}$ denote an estimator of $\sigma_{0}^{2},$ a natural (nominal)\ level $1-\alpha$ confidence interval motivated by the distributional approximation $\sqrt{n}(\hat{\theta}_{n}-\theta_{0})\overset{\cdot}{\sim}\mathcal{N} (0,\hat{\sigma}_{n}^{2})$ is the \textquotedblleft Normal\textquotedblright \ interval given by \[ \mathsf{CI}_{n,1-\alpha}^{\mathtt{N}}=\left[ \hat{\theta}_{n}-\Phi ^{-1}(1-\alpha/2)\hat{\sigma}_{n}/\sqrt{n}~,~\hat{\theta}_{n}-\Phi^{-1} (\alpha/2)\hat{\sigma}_{n}/\sqrt{n}\right] . \] This interval is consistent if $\hat{\sigma}_{n}^{2}$ is consistent. The bootstrap consistency property $\left( \ref{Bootstrap consistency}\right) $ is neither necessary nor sufficient for the \textquotedblleft bootstrap variance consistency\textquotedblright\ property
Following \citet*{Bickel-Freedman_1981_AoS}, one way of ensuring that bootstrap variance consistency is implied by bootstrap consistency is to employ the Mallows metric $d_{2}$ when defining bootstrap consistency. The examples studied herein have the feature that $(\ref{Bootstrap variance consistency})$ can hold even if $\left( \ref{Bootstrap consistency}\right) $ (and therefore also convergence in the Mallows metric) fails. Partly for this reason it seems more attractive (to us at least) to define bootstrap consistency as in $\left( \ref{Bootstrap consistency}\right) ,$ hereby treating bootstrap consistency and bootstrap variance consistency as distinct (i.e., non-nested) properties.
Our discussion of efficiency (or otherwise) of average density estimators $\hat{\theta}_{n}$ will be based on the natural decomposition of the estimation error $\hat{\theta}_{n}-\theta_{0}$ into its bias and \textquotedblleft noise\textquotedblright\ components $\mathbb{E}[\hat{\theta }_{n}]-\theta_{0}$ and $\hat{\theta}_{n}-\mathbb{E}[\hat{\theta}_{n}].\ $If these components satisfy
and
respectively, then $\left( \ref{Efficiency}\right) $ holds. Moreover, if $\left( \ref{Efficiency: Noise condition}\right) $ holds, then the easy-to-interpret bias condition $\left( \ref{Efficiency: Bias condition} \right) $ is necessary and sufficient for $\left( \ref{Efficiency}\right) .$ The latter observation is particularly useful for our purposes, as it turns out that the estimators of interest satisfy $\left( \ref{Efficiency: Noise condition}\right) $ under very mild conditions.
The simplest average density estimator is arguably the kernel-based \textquotedblleft plug-in\textquotedblright\ estimator \[ \hat{\theta}_{n}^{\mathtt{AD}}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{n}(X_{i}), \] where, for some kernel $K$ and some bandwidth $h_{n},$ $\hat{f}_{n}$ denotes the kernel density estimator \[ \hat{f}_{n}\left( x\right) =\frac{1}{n}\sum_{1\leq j\leq n}K_{n} (x-X_{j}),\qquad K_{n}(x)=\frac{1}{h_{n}^{d}}K\left( \frac{x}{h_{n}}\right) . \]
When developing results for $\hat{\theta}_{n}^{\mathtt{AD}}$ and other estimators, we impose the following standard condition on the kernel, in which $\Vert\cdot\Vert_{1}$ denotes the $\ell_{1}$-norm and $u^{l}$ is shorthand for $u_{1}^{l_{1}}\cdots u_{d}^{l_{d}}$ when $u=(u_{1},\ldots,u_{d})^{\prime} \in\mathbb{R}^{d}$ and $l=(l_{1},\ldots,l_{d})^{\prime}\in\mathbb{Z}_{+}^{d}.$
The constant $P$ in Condition K is the order of the kernel. Condition K therefore implies that $K$ is a higher order kernel when $d\geq4.$ As usual, we employ higher order kernels in order to ensure that the magnitude of the smoothing bias of $\hat{f}_{n}$ is sufficiently small.
Under Conditions D and K, the density estimator $\hat{f}_{n}$ is consistent (pointwise) provided the bandwidth satisfies
More importantly, Condition B$^{-}$ implies that the average density estimator $\hat{\theta}_{n}^{\mathtt{AD}}$ satisfies $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D and K.\footnote{Conversely, Condition B$^{-}$ is minimal in the sense that the methods of \citet*{Cattaneo-Crump-Jansson_2014_ET_SmallBW} can be used to show that $\left( \ref{Efficiency: Noise condition}\right) $ can fail if Condition B$^{-}$ is violated.} As a consequence, under Conditions D, K, and B$^{-},$ the estimator $\hat{\theta}_{n}^{\mathtt{AD}}$ is efficient if and only if it satisfies the bias condition $\left( \ref{Efficiency: Bias condition}\right) .$
Using the representation $\theta_{0}=f_{0}^{\Delta}(0),\ $the bias of $\hat{\theta}_{n}^{\mathtt{AD}}$ can be shown to admit the approximation
where the approximation error is of order $n^{-1},$ the first term is a \textquotedblleft leave in\textquotedblright\ bias term (in the terminology of \citet*{Cattaneo-Crump-Jansson_2013_JASA}), and the second term is a smoothing bias term. As previosly mentioned, the function $f_{0}^{\Delta}$ belongs to the H\"{o}lder space $\mathbf{C}^{2s}(\mathbb{R}^{d})$ under Condition D. Using this fact, it follows from a routine calculation (e.g., \citet*[Proposition 1.2]{Tsybakov_2009_Book}) that if Conditions D and K are satisfied and if $h_{n}\rightarrow0,$ then \[ \int_{\mathbb{R}^{d}}K(t)[f_{0}^{\Delta}(h_{n}t)-f_{0}^{\Delta}(0)]dt=O(h_{n} ^{2S}),\qquad S=\min(P/2,s). \]
As a consequence, under Conditions D and K the estimator $\hat{\theta} _{n}^{\mathtt{AD}}$ is efficient provided Condition B$^{-}$ is strengthened to
Existence of a bandwidth sequence satisfying Condition B$^{+}$ requires that the parameter $s$ governing the smoothness of $f_{0}$ satisfies $s>d/2,$ a stronger condition than the (minimal) condition $s>d/4$ included in Condition D.
This shortcoming of $\hat{\theta}_{n}^{\mathtt{AD}}$ is attributable to its leave in bias, as it is the presence of the leave in bias that requires a strengthening of the lower bound on the bandwidth from $nh_{n}^{d} \rightarrow\infty$ to $nh_{n}^{2d}\rightarrow\infty.$ Of course, the leave in bias of $\hat{\theta}_{n}^{\mathtt{AD}}$ is easily avoidable. One option is to employ a kernel satisfying $K(0)=0.$ Recognizing that all standard kernels have $K(0)\neq0,$ a more natural option is to use the \textquotedblleft bias-corrected\textquotedblright\ version of $\hat{\theta}_{n}^{\mathtt{AD}}$ given by \[ \hat{\theta}_{n}^{\mathtt{AD-BC}}=\hat{\theta}_{n}^{\mathtt{AD}}-\frac {K(0)}{nh_{n}^{d}}. \] By construction, the bias of this estimator satisfies \[ \mathbb{E}[\hat{\theta}_{n}^{\mathtt{AD-BC}}]-\theta_{0}\approx\int _{\mathbb{R}^{d}}K(t)[f_{0}^{\Delta}(h_{n}t)-f_{0}^{\Delta}(0)]dt=O(h_{n} ^{2S}), \] so under Conditions D and K the bias condition $\left( \ref{Efficiency: Bias condition}\right) $ is satisfied by $\hat{\theta} _{n}^{\mathtt{AD-BC}}$ provided $nh_{n}^{4S}\rightarrow0,$ implying in turn that $\hat{\theta}_{n}^{\mathtt{AD-BC}}$ is asymptotically efficient under Conditions D and K provided the bandwidth satisfies the following condition, which requires no additional smoothness (as measured by the value of $s$) relative to Condition D.
The leave in bias of $\hat{\theta}_{n}^{\mathtt{AD}}$ is proportional to $1/(nh_{n}^{d}).$ Equipped with only that knowledge, the method of generalized jackknifing constructs a debiased version of $\hat{\theta}_{n}^{\mathtt{AD}}$ as a weighted sum of two (or more) versions of $\hat{\theta}_{n}^{\mathtt{AD} }$ implemented using different values of the bandwidth, where the weights are judiciously chosen to remove the leave in bias. To give the simplest example, let $\hat{\theta}_{n}^{\mathtt{AD}}(h)$ denote the version of $\hat{\theta }_{n}^{\mathtt{AD}}$ associated with the bandwidth $h.$ Then, for any $c\neq1$ the \textquotedblleft generalized jackknife\textquotedblright\ version of $\hat{\theta}_{n}^{\mathtt{AD}}$ obtained by combining $\hat{\theta} _{n}^{\mathtt{AD}}=\hat{\theta}_{n}^{\mathtt{AD}}(h_{n})$ and $\hat{\theta }_{n}^{\mathtt{AD}}(ch_{n})$ is given by \[ \hat{\theta}_{n}^{\mathtt{AD-GJ}}=\frac{1}{1-c^{d}}\hat{\theta}_{n} ^{\mathtt{AD}}-\frac{c^{d}}{1-c^{d}}\hat{\theta}_{n}^{\mathtt{AD}}(ch_{n}). \] Like $\hat{\theta}_{n}^{\mathtt{AD}},$ the estimator $\hat{\theta} _{n}^{\mathtt{AD-GJ}}$ satisfies $\left( \ref{Efficiency: Noise condition} \right) $ under Conditions D, K, and B$^{-}$. Moreover, because \[ \mathbb{E}[\hat{\theta}_{n}^{\mathtt{AD}}(ch_{n})]-\theta_{0}\approx\frac {1}{c^{d}}\frac{K(0)}{nh_{n}^{d}}+\int_{\mathbb{R}^{d}}K(t)[f_{0}^{\Delta }(ch_{n}t)-f_{0}^{\Delta}(0)]dt, \] the bias condition $\left( \ref{Efficiency: Bias condition}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{AD-GJ}}$ under Condition B.
Finally, as its name suggests, the leave in bias can also be avoided by employing \textquotedblleft leave out\textquotedblright\ estimators of $f_{0}.$ A generic average density estimator based on leave out density estimators is of the form \[ \hat{\theta}_{n}^{\mathtt{AD-LO}}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{i,n}^{\mathtt{LO}}(X_{i}), \] where $\hat{f}_{i,n}^{\mathtt{LO}}$ is a kernel density estimator constructed using observations belonging to a set that does not include $X_{i}.$ Relative to $\hat{\theta}_{n}^{\mathtt{AD-BC}}$ and $_{n}^{\mathtt{AD-GJ}},$ an attractive feature of $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ is that it can be constructed without knowledge of the functional form of the leave-in bias. For concreteness, we shall develop results for $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ only in the (leading) special case where the sample $X_{1},\ldots,X_{n}$ is partitioned into $B_{n}\in\{2,\ldots,n\}$ disjoint blocks of (approximately) equal size and $\hat{f}_{i,n}^{\mathtt{LO}}$ is constructed using observations from all blocks except the one to which the $i $th observation belongs. To be specific, we assume that $\hat{f}_{i,n}^{\mathtt{LO}}$ is of the form \[ \hat{f}_{i,n}^{\mathtt{LO}}(x)=\sum_{1\leq j\leq n}w_{ij,n}K_{n} (x-X_{j}),\qquad w_{ij,n}=\frac{ {\rm 1\hspace*{-0.4ex}\rule{0.1ex}{1.52ex}\hspace*{0.2ex}} (\left\lceil iB_{n}/n\right\rceil \neq\left\lceil jB_{n}/n\right\rceil )} {\sum_{1\leq k\leq n} {\rm 1\hspace*{-0.4ex}\rule{0.1ex}{1.52ex}\hspace*{0.2ex}} (\left\lceil iB_{n}/n\right\rceil \neq\left\lceil kB_{n}/n\right\rceil )}. \]
When $B_{n}=n,$ $\hat{f}_{i,n}^{\mathtt{LO}}$ is the $i$th \textquotedblleft leave-one-out\textquotedblright\ estimator of $f_{0}$ and the estimator $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ reduces to the estimator introduced in \citet*{Hall-Marron_1987_SPL} and further studied by \citet*{Gine-Nickl_2008_Bernoulli} (among many others). At the opposite extreme, when $B_{n}$ is kept fixed, the estimator $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ is a \textquotedblleft cross-fit\textquotedblright\ estimator (using an $B_{n}$-fold non-random partition of $\{1,\ldots,n\}$) in the terminology of \citet*{Newey-Robins_2018_CrossFitting}.
Regardless of the choice of $B_{n},$ under Conditions D, K, and B$^{-},$ the estimator $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ is similar to $\hat{\theta} _{n}^{\mathtt{AD-BC}}$ and $\hat{\theta}_{n}^{\mathtt{AD-GJ}}$ insofar as it satisfies $\left( \ref{Efficiency: Noise condition}\right) $ and has \[ \mathbb{E}[\hat{\theta}_{n}^{\mathtt{AD-LO}}]-\theta_{0}\approx\int _{\mathbb{R}^{d}}K(t)[f_{0}^{\Delta}(h_{n}t)-f_{0}^{\Delta}(0)]dt=O(h_{n} ^{2S}), \] implying in particular that $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ is asymptotically efficient under Conditions D, K and B.
The following result collects and summarizes the main findings of this section.
\noindentRemark. Because $\hat{\theta}_{n}^{\mathtt{AD}}$ is linear functional of $\hat{f}_{n},$ the generalized jackknife estimator $\hat{\theta }_{n}^{\mathtt{AD-GJ}}$ can be interpreted as a version of the plug-in estimator $\hat{\theta}_{n}^{\mathtt{AD}}$ based on a modified kernel: Defining \[ K^{\mathtt{GJ}}(x)=\frac{1}{1-c^{d}}\left[ K(x)-K\left( \frac{x}{c}\right) \right] , \] we have \[ \hat{\theta}_{n}^{\mathtt{AD-GJ}}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{n}^{\mathtt{GJ}}(X_{i}), \] where \[ \hat{f}_{n}^{\mathtt{AD-GJ}}\left( x\right) =\frac{1}{n}\sum_{1\leq j\leq n}K_{n}^{\mathtt{GJ}}(x-X_{j}),\qquad K_{n}^{\mathtt{GJ}}(x)=\frac{1} {h_{n}^{d}}K^{\mathtt{GJ}}\left( \frac{x}{h_{n}}\right) . \] The modified kernel satisfies $K^{\mathtt{GJ}}(0)=0,$ so this interpretation provides an explanation of the fact that $\hat{\theta}_{n}^{\mathtt{AD-GJ}}$ satisfies $\left( \ref{Efficiency: Bias condition}\right) $ under Condition B. A similar interpretation is not available for generalized jackknife versions of estimators that are nonlinear functionals of $\hat{f}_{n};$ examples of such estimators are given by $\hat{\theta}_{n}^{\mathtt{ISD-GJ}}$ and $\hat{\theta}_{n}^{\mathtt{LR-GJ}}$ studied in Section (ref).
Letting $X_{1,n}^{\ast},\ldots,X_{n,n}^{\ast}$ denote a random sample from the empirical distribution of $X_{1},\ldots,X_{n},$ the natural bootstrap analogs of the estimators studied in the previous section are given by \[ \hat{\theta}_{n}^{\mathtt{AD,}\ast}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{n}^{\ast}(X_{i,n}^{\ast}),\qquad\hat{f}_{n}^{\ast}\left( x\right) =\frac{1}{n}\sum_{1\leq j\leq n}K_{n}(x-X_{j,n}^{\ast}), \] \[ \hat{\theta}_{n}^{\mathtt{AD-BC,}\ast}=\hat{\theta}_{n}^{\mathtt{AD,}\ast }-\frac{K(0)}{nh_{n}^{d}}, \] \[ \hat{\theta}_{n}^{\mathtt{AD-GJ,}\ast}=\frac{1}{1-c^{d}}\hat{\theta} _{n}^{\mathtt{AD,}\ast}-\frac{c^{d}}{1-c^{d}}\hat{\theta}_{n}^{\mathtt{AD,} \ast}(ch_{n}), \] and \[ \hat{\theta}_{n}^{\mathtt{AD-LO,}\ast}=\frac{1}{n}\sum_{1\leq i\leq n}\hat {f}_{i,n}^{\mathtt{LO,}\ast}(X_{i,n}^{\ast}),\qquad\hat{f}_{i,n} ^{\mathtt{LO,}\ast}(x)=\sum_{1\leq j\leq n}w_{ij,n}K_{n}(x-X_{j,n}^{\ast}), \] respectively, where $\hat{\theta}_{n}^{\mathtt{AD,}\ast}(ch_{n})$ denotes the version of $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ associated with the bandwidth $ch_{n}.$ The main goal of this section is to explore the extent to which these estimators enjoy the bootstrap consistency property $\left( \ref{Bootstrap consistency}\right) $ under Conditions D, K, and B.
If $\hat{\theta}_{n}$ is efficient in the sense that it satisfies $\left( \ref{Efficiency}\right) ,$ then $\sqrt{n}(\hat{\theta}_{n}-\theta _{0})\rightsquigarrow\mathcal{N}(0,\sigma_{0}^{2}),$ implying in particular that the bootstrap consistency property $\left( \ref{Bootstrap consistency} \right) $ admits the following characterization:
where $\rightsquigarrow_{\mathbb{P}}$ denotes conditional weak convergence in probability.
Similarly to the analysis of the previous section, it seems natural to base verification of $\left( \ref{Bootstrap consistency: Characterization under efficiency}\right) $ on a decomposition of the bootstrap estimation error $\hat{\theta}_{n}^{\ast} -\hat{\theta}_{n}$ into its bias and noise\ components $\mathbb{E}_{n}^{\ast }[\hat{\theta}_{n}^{\ast}]-\hat{\theta}_{n} $ and $\hat{\theta}_{n}^{\ast }-\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\ast}],$ where $\mathbb{E}_{n} ^{\ast}[\cdot]=\mathbb{E}[\cdot|X_{1},\ldots,X_{n}].$ The resulting sufficient condition for $\left( \ref{Bootstrap consistency: Characterization under efficiency}\right) $ is given by the pair
and
where $\left( \ref{Bootstrap consistency: Bias condition under efficiency} \right) $ is the natural bootstrap analog of $\left( \ref{Efficiency: Bias condition}\right) ,$ $\left( \ref{Bootstrap consistency: Noise condition}\right) $ is a bootstrap version of the main distributional implication of $\left( \ref{Efficiency: Noise condition}\right) ,$ and where $\left( \ref{Bootstrap consistency: Bias condition under efficiency}\right) $ is necessary and sufficient for $\left( \ref{Bootstrap consistency: Characterization under efficiency}\right) $ when $\left( \ref{Bootstrap consistency: Noise condition}\right) $ holds.
In perfect analogy with $\left( \ref{Efficiency: Noise condition}\right) ,$ it turns out that $\left( \ref{Bootstrap consistency: Noise condition} \right) $ holds under very mild bandwidth conditions. Indeed, under Conditions D and K, the estimators $\hat{\theta}_{n}^{\mathtt{AD,}\ast},$ $\hat{\theta}_{n}^{\mathtt{AD-BC,}\ast},$ $\hat{\theta}_{n}^{\mathtt{AD-GJ,} \ast},$ and $\hat{\theta}_{n}^{\mathtt{AD-LO,}\ast}$ all satisfy $\left( \ref{Bootstrap consistency: Noise condition}\right) $ whenever Condition B$^{-}$ holds.\footnote{Conversely, Condition B$^{-}$ is minimal in the sense that the methods of \citet*{Cattaneo-Crump-Jansson_2014_ET_Bootstrap} can be used to show that $\left( \ref{Bootstrap consistency: Noise condition} \right) $ can fail if Condition B$^{-}$ is violated.} As a consequence, the question once again becomes whether the estimators have biases that are sufficiently small. Under Conditions D, K, and B$^{-}$, the bootstrap bias of $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ satisfies
Therefore, the bias condition $\left( \ref{Bootstrap consistency: Bias condition under efficiency}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ provided $nh_{n} ^{2d}\rightarrow\infty.$ In other words, $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ satisfies $\left( \ref{Bootstrap consistency}\right) $ (and therefore also $\left( \ref{Consistency of percentile interval}\right) $ and $\left( \ref{Efficiency of percentile interval}\right) $) under Conditions D, K, and B$^{+}$.
More surprisingly, perhaps, although the estimator $\hat{\theta} _{n}^{\mathtt{AD-BC}}$ is efficient under Conditions D, K, and B, stronger conditions are required for its bootstrap analog $\hat{\theta}_{n} ^{\mathtt{AD-BC,}\ast}$ to satisfy $\left( \ref{Bootstrap consistency} \right) .$ This is so because
under Conditions D, K, and B. A similar remark applies to $\hat{\theta} _{n}^{\mathtt{AD-LO}},$ as its bootstrap analog satisfies \[ \mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{AD-LO,}\ast}]-\hat{\theta} _{n}^{\mathtt{AD-LO}}=\hat{\theta}_{n}^{\mathtt{AD}}-\hat{\theta} _{n}^{\mathtt{AD-LO}}=\frac{K(0)}{nh_{n}^{d}}+o_{\mathbb{P}}(n^{-1/2}) \] under Conditions D, K, and B.
On the other hand, because \[ \mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{AD,}\ast}(ch_{n})]-\hat {\theta}_{n}^{\mathtt{AD}}(ch_{n})=\frac{1}{c^{d}}\frac{K(0)}{nh_{n}^{d} }-\frac{1}{n}\hat{\theta}_{n}^{\mathtt{AD}}(ch_{n})=\frac{1}{c^{d}}\frac {K(0)}{nh_{n}^{d}}+O_{\mathbb{P}}(n^{-1}), \] the bootstrap analog of $\hat{\theta}_{n}^{\mathtt{AD-GJ}}$ satisfies \[ \mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{AD-GJ,}\ast}]-\hat{\theta} _{n}^{\mathtt{AD-GJ}}=O_{\mathbb{P}}(n^{-1}), \] so this estimator satisfies $\left( \ref{Bootstrap consistency}\right) $ under Conditions D, K, and B.
It turns out that $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$\ satisfies $\left( \ref{Bootstrap consistency}\right) ,$ $\left( \ref{Consistency of percentile interval}\right) ,$ and $\left( \ref{Efficiency of percentile interval}\right) $ under conditions that are weaker than the conditions under which $\hat{\theta}_{n}^{\mathtt{AD}}$ is efficient. In generic notation, suppose the estimators $\hat{\theta}_{n}$ and $\hat{\theta}_{n}^{\ast}$ satisfy $\left( \ref{Efficiency: Noise condition} \right) $ and $\left( \ref{Bootstrap consistency: Noise condition}\right) ,$ respectively. Then $\left( \ref{Bootstrap consistency}\right) $ is still sufficient for $\left( \ref{Consistency of percentile interval}\right) ,$ and $\left( \ref{Efficiency of percentile interval}\right) $ to hold. Moreover, as also observed by \citet*{Cattaneo-Jansson_2018_ECMA}, the bootstrap consistency condition $\left( \ref{Bootstrap consistency}\right) $ itself is satisfied under the following generalization of the bias conditions $\left( \ref{Efficiency: Bias condition}\right) $ and $\left( \ref{Bootstrap consistency: Bias condition under efficiency}\right) :$
Now, as discussed above, the estimators $\hat{\theta}_{n}^{\mathtt{AD}}$ and $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ satisfy $\left( \ref{Efficiency: Noise condition}\right) $ and $\left( \ref{Bootstrap consistency: Noise condition}\right) ,$ respectively, under Conditions D, K, and B. Under the same conditions, it follows from $\left( \ref{Plug-In AD estimator: Bias}\right) $ and $\left( \ref{Plug-In AD estimator: Bootstrap bias}\right) $\ that $\left( \ref{Bootstrap consistency: Bias condition}\right) $ is satisfied.
The following result collects and summarizes the main findings of this section.
Comparing Theorems (ref) and (ref), we see that efficiency is neither necessary nor sufficient for bootstrap consistency. In fact, the results indicate that there can be a tension between efficiency and bootstrap consistency in semiparametric settings. What seems most noteworthy to us is that whereas \textquotedblleft debiased\textquotedblright\ estimators such as $\hat{\theta}_{n}^{\mathtt{AD-BC}}$ and $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ may appear to be superior to the simple plug-in estimator $\hat{\theta} _{n}^{\mathtt{AD}}$ insofar as they achieve efficiency under weaker (indeed, minimal) conditions, the ranking gets reversed when the estimators are looked at through the lens of the bootstrap. As pointed out by \citet*{Chen-Linton-vanKeilegom_2003_ECMA} and \citet*{Cheng-Huang_2010_AoS}, bootstrap-based inference is particularly attractive in semiparametric settings. The results above demonstrate by example that efficiency-based rankings of estimators can be quite misleading\ in cases where construction of an estimator is simply a means to the end of conducting bootstrap-based inference.
As conjectured by the co-editor, the estimators $\hat{\theta}_{n} ^{\mathtt{AD}},$ $\hat{\theta}_{n}^{\mathtt{AD-BC}},$ $\hat{\theta} _{n}^{\mathtt{AD-GJ}}, $ and $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ can all be shown to satisfy the bootstrap variance consistency property $(\ref{Bootstrap variance consistency})$ under Conditions D, K, and B. The estimators $\hat{\theta}_{n}^{\mathtt{AD-BC}},$ $\hat{\theta}_{n} ^{\mathtt{AD-GJ}},$ and $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ therefore enjoy the property that the intervals $\mathsf{CI}_{n,1-\alpha}^{\mathtt{N}}$ based on the bootstrap variance estimator are consistent (indeed, efficient) under Conditions D, K, and B.
An important source of the bootstrap consistency result for $\hat{\theta} _{n}^{\mathtt{AD}}$ is the ability of the bootstrap to automatically perform a bias correction when approximating the distribution of $\hat{\theta} _{n}^{\mathtt{AD}}-\theta_{0}.$ The same mechanism can be exploited for estimation purposes: Setting $\alpha=1,$ the interval $\mathsf{CI} _{n,1-\alpha}^{\mathtt{P}}$ becomes a singleton and can therefore be interpreted as a bootstrap-based estimator of $\theta_{0}.$ As a by-product of our results about $\hat{\theta}_{n}^{\mathtt{AD}},$ it can be shown that the resulting estimator \[ 2\hat{\theta}_{n}^{\mathtt{AD}}-\inf\{q\in\mathbb{R}:\mathbb{P}_{n}^{\ast }[\hat{\theta}_{n}^{\mathtt{AD,}\ast}\leq q]\geq1/2\}, \] is efficient under Conditions D, K, and B.
The bootstrap analog of $\hat{\theta}_{n}^{\mathtt{AD}}$ employs a density estimator $\hat{f}_{n}^{\ast}$ that uses the same bandwidth $h_{n}$ as is used when constructing $\hat{f}_{n}.$ Doing so is important for the purposes of obtaining the bootstrap consistency result for $\hat{\theta}_{n}^{\mathtt{AD} }.$ Indeed, if $\hat{f}_{n}^{\ast}$ were defined using a possibly different bandwidth $h_{n}^{\ast}$ (say), then the bootstrap consistency result under Condition B can fail unless $h_{n}^{\ast}/h_{n}\rightarrow_{\mathbb{P}}1.$ On the other hand, the flavor of the bootstrap results about $\hat{\theta} _{n}^{\mathtt{AD-BC}},$ $\hat{\theta}_{n}^{\mathtt{AD-LO}},$ and $\hat{\theta }_{n}^{\mathtt{AD-GJ}}$ does not change if a different bandwidth is used when defining their bootstrap analogs.
In light of Theorem (ref) it is of interest to construct bootstrap-based approximations to the distributions of $\hat{\theta }_{n}^{\mathtt{AD-BC}}$ and $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ that are consistent under Conditions D, K, and B. In generic notation, suppose $\hat{\theta}_{n}=\hat{\theta}_{n}(X_{1},\ldots,X_{n})$ is the estimator whose distribution we seek to approximate. One option is to find an estimator $\tilde{\theta}_{n}=\tilde{\theta}_{n}(X_{1},\ldots,X_{n})$ (say) whose natural bootstrap analog $\tilde{\theta}_{n}^{\ast}=\tilde{\theta}_{n} (X_{1,n}^{\ast},\ldots,X_{n,n}^{\ast})$ satisfies
As we shall see, both $\hat{\theta}_{n}^{\mathtt{AD-BC}}$ and $\hat{\theta }_{n}^{\mathtt{AD-LO}}$ lend themselves well to a construction of this type. Nevertheless, in some circumstances it may be equally (if not more) attractive to achieve consistency by finding a bootstrap probability measure $\mathbb{P}_{n}^{\star}$ (say) governing the distribution of $X_{1,n}^{\star },\ldots,X_{n,n}^{\star}$ such that $\hat{\theta}_{n}^{\star}=\hat{\theta} _{n}(X_{1,n}^{\star},\ldots,X_{n,n}^{\star})$ satisfies
A construction of this type turns out to be useful in the case of the cross-fit version of $\hat{\theta}_{n}^{\mathtt{AD-LO}}.$
First, consider the problem of approximating the distribution of $\hat{\theta }_{n}^{\mathtt{AD-BC}}.$ It follows from $\left( \ref{Bias-Corrected AD estimator: Bootstrap bias}\right) $ that a bias-corrected version of $\hat{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ is given by \[ \tilde{\theta}_{n}^{\mathtt{AD-BC,}\ast}=\hat{\theta}_{n}^{\mathtt{AD-BC,} \ast}-\frac{K(0)}{nh_{n}^{d}}. \] Rather than showing $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ by analyzing $\tilde{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ directly, we find it more insightful to obtain the consistency result by means of an argument which highlights and exploits the relationship between $\tilde{\theta} _{n}^{\mathtt{AD-BC,}\ast}$ and $\hat{\theta}_{n}^{\mathtt{AD,}\ast}.$ Heuristically, $\tilde{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ \textquotedblleft should\textquotedblright\ satisfy $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ under Conditions D, K, and B because the percentile interval associated with $\tilde{\theta }_{n}^{\mathtt{AD-BC,}\ast}$ is identical to the percentile interval associated with $\hat{\theta}_{n}^{\mathtt{AD,}\ast}.$\footnote{In generic notation, the percentile interval associated with an estimator $\tilde{\theta }_{n}^{\ast}$ is given by \[ \widetilde{\mathsf{CI}}_{n,1-\alpha}=\left[ \hat{\theta}_{n}-\tilde {q}_{n,1-\alpha/2}^{\ast}~,~\hat{\theta}_{n}-\tilde{q}_{n,\alpha/2}^{\ast }\right] ,\qquad\tilde{q}_{n,a}^{\ast}=\inf\{q\in\mathbb{R}:\mathbb{P} _{n}^{\ast}[(\tilde{\theta}_{n}^{\ast}-\hat{\theta}_{n})\leq q]\geq a\}. \] } These heuristics can be made rigorous with the help of the equality
which implies in particular that $\tilde{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ if and only if $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ satisfies $\left( \ref{Bootstrap consistency}\right) .$ As a consequence, the fact $\tilde{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ under Conditions D, K, and B is simply a restatement of the bootstrap consistency result for $\hat{\theta}_{n}^{\mathtt{AD,}\ast}.$
Turning next to $\hat{\theta}_{n}^{\mathtt{AD-LO}},$ our preferred modification of this estimator is motivated by the observation that \[ \mathbb{P}[\tilde{f}_{i,n}^{\mathtt{LO}}(X_{i})=\hat{f}_{i,n}^{\mathtt{LO} }(X_{i})]=1, \] where \[ \tilde{f}_{i,n}^{\mathtt{LO}}(x)=\sum_{1\leq j\leq n}w_{ij,n}\tilde{K} _{n}(x-X_{j}),\qquad\tilde{K}_{n}(x)= {\rm 1\hspace*{-0.4ex}\rule{0.1ex}{1.52ex}\hspace*{0.2ex}} (x\neq0)K_{n}(x). \] An immediate implication of this observation is that \[ \mathbb{P}[\tilde{\theta}_{n}^{\mathtt{AD-LO}}=\hat{\theta}_{n} ^{\mathtt{AD-LO}}]=1,\qquad\tilde{\theta}_{n}^{\mathtt{AD-LO}}=\frac{1}{n} \sum_{1\leq i\leq n}\tilde{f}_{i,n}^{\mathtt{LO}}(X_{i}). \] Nevertheless, unlike $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ itself, the modification $\tilde{\theta}_{n}^{\mathtt{AD-LO}}$ has a natural bootstrap analog \[ \tilde{\theta}_{n}^{\mathtt{AD-LO,}\ast}=\frac{1}{n}\sum_{1\leq i\leq n} \tilde{f}_{i,n}^{\mathtt{LO,}\ast}(X_{i,n}^{\ast}),\qquad\tilde{f} _{i,n}^{\mathtt{LO,}\ast}(x)=\sum_{1\leq j\leq n}w_{ij,n}\tilde{K} _{n}(x-X_{j,n}^{\ast}), \] whose bias is small: Under Conditions D, K, and B, \[ \mathbb{E}_{n}^{\ast}[\tilde{\theta}_{n}^{\mathtt{AD-LO,}\ast}]=\frac{1} {n}\sum_{1\leq i\leq n}\tilde{f}_{n}^{\mathtt{LO}}(X_{i})=\tilde{\theta} _{n}^{\mathtt{AD-LO}}+o_{\mathbb{P}}(n^{-1/2}),\qquad\tilde{f}_{n} ^{\mathtt{LO}}(x)=\frac{1}{n}\sum_{1\leq j\leq n}\tilde{K}_{n}(x-X_{j}). \] In fact, it can be shown that $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ is satisfied by $\tilde{\theta}_{n}^{\mathtt{AD-LO,}\ast}$ under Conditions D, K, and B.
For cross-fit estimators, an arguably more attractive option is to construct a bootstrap-based distributional approximation which employs a bootstrap probability measure that is itself of cross-fit (i.e., split sample) type. To illustrate the idea, we consider the simplest special case. When $B_{n}=2, $ the estimator $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ reduces to \[ \hat{\theta}_{n}^{\mathtt{AD-CF}}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{i,n}^{\mathtt{CF}}(X_{i}), \] where \[ \hat{f}_{i,n}^{\mathtt{CF}}(x)=\left\{
\right. . \] The $B_{n}=2$ version of the \textquotedblleft cross-fit bootstrap\textquotedblright\ is defined as follows. Conditional on $X_{1},\ldots,X_{n},$ let $X_{1,n}^{\star},\ldots,X_{n,n}^{\star}$ be mutually independent with $X_{1,n}^{\star},\ldots,X_{\left\lfloor n/2\right\rfloor ,n}^{\star}$ being a random sample from the empirical distribution of $X_{1},\ldots,X_{\left\lfloor n/2\right\rfloor }$ and $X_{\left\lfloor n/2\right\rfloor +1,n}^{\star},\ldots,X_{n,n}^{\star}$ being a random sample from the empirical distribution of $X_{\left\lfloor n/2\right\rfloor +1},\ldots,X_{n}.$ Then, \[ \hat{\theta}_{n}^{\mathtt{AD-CF,}\star}=\frac{1}{n}\sum_{1\leq i\leq n}\hat {f}_{i,n}^{\mathtt{CF,}\star}(X_{i,n}^{\star}) \] is the corresponding cross-fit bootstrap\ version of $\hat{\theta} _{n}^{\mathtt{AD-CF}},$ where \[ \hat{f}_{i,n}^{\mathtt{CF,}\star}(x)=\left\{
\right. . \] The bootstrap distribution of $\hat{\theta}_{n}^{\mathtt{AD-CF,}\star}$ is correctly centered in the sense that $\mathbb{E}_{n}^{\star}[\hat{\theta} _{n}^{\mathtt{AD-CF,}\star}]=\hat{\theta}_{n}^{\mathtt{AD-CF}},$ where $\mathbb{E}_{n}^{\star}[\cdot]$ denotes the expected value computed under the cross-fit bootstrap distribution. In fact, the bootstrap distribution satisfies $\left( \ref{Bootstrap consistency: Alternative measure}\right) $ under Conditions D, K, and B.
As pointed out by a referee, yet another way of achieving consistency on the part of a bootstrap-based distributional approximation is to center the distribution of $\hat{\theta}_{n}^{\ast}$ at an estimator $\tilde{\theta}_{n}$ satisfying
Because the estimators under consideration here all satisfiy $\left( \ref{Efficiency: Noise condition}\right) $ and $\left( \ref{Bootstrap consistency: Noise condition}\right) ,$ the following analog of $\left( \ref{Bootstrap consistency: Bias condition}\right) $ is sufficient for $\left( \ref{Bootstrap consistency: Alternative centering} \right) :$ \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\ast}]-\tilde{\theta} _{n})=\sqrt{n}(\mathbb{E}[\hat{\theta}_{n}]-\theta_{0})+o_{\mathbb{P}}(1). \] As already mentioned in connection with $\left( \ref{Bootstrap consistency: Bias condition}\right) ,$ the displayed condition satisfied by $\tilde{\theta}_{n}=\hat{\theta}_{n}$ in the case of $\hat {\theta}_{n}^{\mathtt{AD}}.$ For the other estimators (i.e., for $\hat{\theta }_{n}^{\mathtt{AD-BC}},$ $\hat{\theta}_{n}^{\mathtt{AD-GJ}},$ and $\hat {\theta}_{n}^{\mathtt{AD-LO}}$), because they satisfy the bias condition $\left( \ref{Efficiency: Bias condition}\right) ,$ the displayed condition is satisfied by $\tilde{\theta}_{n}=\mathbb{E}_{n}^{\ast}[\hat{\theta} _{n}^{\ast}].$
This section considers two alternative classes of estimators. The first class is motivated by the integrated squared density representation \[ \theta_{0}=\int_{\mathbb{R}^{d}}f_{0}\left( x\right) ^{2}dx, \] an interesting feature of which is that it involves a nonlinear functional of $f_{0}.$ The second class is motivated by the representation \[ \theta_{0}=2\mathbb{E}[f_{0}(X)]-\int_{\mathbb{R}^{d}}f_{0}\left( x\right) ^{2}dx, \] an interesting feature of which is that it is \textquotedblleft locally robust\textquotedblright/\textquotedblleft Neyman orthogonal\textquotedblright \ (in the terminology of \citet*{Chernozhukov-Escanciano-Ichimura-Newey-Robins_2020_LocallyRobust}).
A kernel-based plug-in integrated squared density estimator is \[ \hat{\theta}_{n}^{\mathtt{ISD}}=\int_{\mathbb{R}^{d}}\hat{f}_{n}\left( x\right) ^{2}dx. \] Like $\hat{\theta}_{n}^{\mathtt{AD}},$ this estimator has a (potentially) nonnegligible bias: Under Conditions D, K, and B, \[ \mathbb{E}[\hat{\theta}_{n}^{\mathtt{ISD}}]-\theta_{0}=\frac{\int _{\mathbb{R}^{d}}K(u)^{2}du}{nh_{n}^{d}}+o(n^{-1/2}), \] where the first term is a \textquotedblleft nonlinearity\textquotedblright \ bias term (in the terminology of \citet*{Cattaneo-Crump-Jansson_2013_JASA}) attributable to the fact that $\hat{\theta}_{n}^{\mathtt{ISD}}$ is a nonlinear functional of $\hat{f}_{n}.$
The nonlinearity bias of $\hat{\theta}_{n}^{\mathtt{ISD}}$ is easily avoidable, a simple bias-corrected version of $\hat{\theta}_{n}^{\mathtt{ISD} }$ being \[ \hat{\theta}_{n}^{\mathtt{ISD-BC}}=\hat{\theta}_{n}^{\mathtt{ISD}}-\frac {\int_{\mathbb{R}^{d}}K(u)^{2}du}{nh_{n}^{d}}. \] Similarly, because the nonlinearity bias of $\hat{\theta}_{n}^{\mathtt{ISD}}$ is proportional to $1/(nh_{n}^{d}),$ the following generalized jackknife version of $\hat{\theta}_{n}^{\mathtt{ISD}}$ is an efficient estimator of $\theta_{0}:$ \[ \hat{\theta}_{n}^{\mathtt{ISD-GJ}}=\frac{1}{1-c^{d}}\hat{\theta} _{n}^{\mathtt{ISD}}-\frac{c^{d}}{1-c^{d}}\hat{\theta}_{n}^{\mathtt{ISD} }(ch_{n}), \] where $c\neq1$ is a user-chosen constant and where $\hat{\theta} _{n}^{\mathtt{ISD}}(ch_{n})$ denotes the version of $\hat{\theta} _{n}^{\mathtt{ISD}}$ associated with the bandwidth $ch_{n}.$
On the other hand, because the source of the nonlinearity bias of $\hat {\theta}_{n}^{\mathtt{ISD}}$ is different from the source of the leave in bias of $\hat{\theta}_{n}^{\mathtt{AD}},$ there is no particular reason to expect leave out estimators of the form \[ \hat{\theta}_{n}^{\mathtt{ISD-LO}}=\frac{1}{n}\sum_{1\leq i\leq n} \int_{\mathbb{R}^{d}}\hat{f}_{i,n}^{\mathtt{LO}}(x)^{2}dx \] to have favorable bias properties. Indeed, under Conditions D, K, and B and assuming $B_{n}$ is proportional to $n,$ we have:\footnote{More generally (i.e., whether or not $B_{n}$ is proportional to $n$), it is shown in the proof of Theorem (ref) that the bias expansion is of the form \[ \mathbb{E}[\hat{\theta}_{n}^{\mathtt{ISD-LO}}]-\theta_{0}=\eta_{n}\frac {\int_{\mathbb{R}^{d}}K(u)^{2}du}{nh_{n}^{d}}+o(n^{-1/2}), \] where $\eta_{n}\geq1$ is bounded.}
so the nonlinearity bias of $\hat{\theta}_{n}^{\mathtt{ISD-LO}}$ is nonnegligible (and no smaller than that of $\hat{\theta}_{n}^{\mathtt{ISD}}$).
Nevertheless, because $\theta_{0}$ is a quadratic functional of $f_{0},$ the method of \textquotedblleft doubly cross-fitting\textquotedblright\ (in the terminology of \citet*{Newey-Robins_2018_CrossFitting}) can be used to construct an estimator which is free of nonlinearity bias and can be implemented without knowledge of the functional form of the nonlinearity bias. One such estimator is \[ \hat{\theta}_{n}^{\mathtt{ISD-DCF}}=\int_{\mathbb{R}^{d}}\hat{f} _{1,n}^{\mathtt{CF}}\left( x\right) \hat{f}_{n,n}^{\mathtt{CF}}\left( x\right) dx, \] whose bias turns out to be negligible under Conditions D, K, and B.
Under Conditions D, K, and B$^{-}$, the estimators $\hat{\theta} _{n}^{\mathtt{ISD}},$ $\hat{\theta}_{n}^{\mathtt{ISD-BC}},$ $\hat{\theta} _{n}^{\mathtt{ISD-GJ}},$ $\hat{\theta}_{n}^{\mathtt{ISD-LO}},$ and $\hat{\theta}_{n}^{\mathtt{ISD-DCF}}$ all satisfy $\left( \ref{Efficiency: Noise condition}\right) .$ As a consequence, we obtain the following integrated squared density counterpart of Theorem (ref).
An integrated squared density counterpart of Theorem (ref) is also available. Under Conditions D, K, and B, if $\hat{\theta}_{n}\in\{\hat{\theta}_{n}^{\mathtt{ISD}},\hat {\theta}_{n}^{\mathtt{ISD-BC}},\hat{\theta}_{n}^{\mathtt{ISD-LO}},\hat{\theta }_{n}^{\mathtt{ISD-DCF}}\},$ then its bootstrap analog satisfies $\left( \ref{Bootstrap consistency: Noise condition}\right) $ and has a bias of the form \[ \mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\ast}]-\hat{\theta}_{n}=\frac {\int_{\mathbb{R}^{d}}K(u)^{2}du}{nh_{n}^{d}}+o_{\mathbb{P}}(n^{-1/2}), \] so $\left( \ref{Bootstrap consistency}\right) $ is satisfied if (and only if) \[ \mathbb{E}[\hat{\theta}_{n}]-\theta_{0}=\frac{\int_{\mathbb{R}^{d}}K(u)^{2} du}{nh_{n}^{d}}+o(n^{-1/2}). \] \newline The latter condition is satisfied by $\hat{\theta}_{n}^{\mathtt{ISD} },$ but violated by $\hat{\theta}_{n}^{\mathtt{ISD-BC}}$ and $\hat{\theta} _{n}^{\mathtt{ISD-DCF}}.$ In the case of $\hat{\theta}_{n}^{\mathtt{ISD-LO}},$ it follows from $\left( \ref{Leave-Out ISD estimator: Bias}\right) $ that the condition is satisfied when $B_{n}=n$ (i.e., when $\hat{\theta} _{n}^{\mathtt{ISD-LO}}$ is a leave-one-out estimator), but violated when $B_{n}$ is fixed (i.e., when $\hat{\theta}_{n}^{\mathtt{ISD-LO}}$ is a cross-fit estimator).
In important respects, the results reported in Theorems (ref) and (ref) are in qualitative agreement with those reported in Theorems (ref) and (ref). In particular, we find that in spite of being inefficient the simple plug-in estimator achieves bootstrap consistency under conditions that are weaker than those required for efficient estimators to achieve bootstrap consistency. The most notable difference between the integrated squared density and average derivative estimators is probably that in the case of integrated squared density estimators, the cross-fit estimator is demonstrably worse than the plug-in estimator, satisfying neither $\left( \ref{Efficiency}\right) $ nor $\left( \ref{Bootstrap consistency}\right) .$
As was the case with the average density estimators, the integrated squared density estimators can all be shown to satisfy the bootstrap variance consistency property $(\ref{Bootstrap variance consistency})$ under Conditions D, K, and B. The estimators $\hat{\theta}_{n}^{\mathtt{ISD-BC}},$ $\hat {\theta}_{n}^{\mathtt{ISD-GJ}},$ and $\hat{\theta}_{n}^{\mathtt{ISD-DCF}}$ therefore enjoy the property that the intervals $\mathsf{CI}_{n,1-\alpha }^{\mathtt{N}}$ based on the bootstrap variance estimator are consistent (indeed, efficient) under Conditions D, K, and B.
For completeness, we conclude this subsection by briefly discussing integrated squared density versions of $\left( \ref{Bootstrap consistency: Alternative estimator}\right) ,$ $\left( \ref{Bootstrap consistency: Alternative measure}\right) $, and $\left( \ref{Bootstrap consistency: Alternative centering}\right) .$ In what follows, suppose Conditions D, K, and B are satisfied. A bias-corrected version of $\hat{\theta}_{n}^{\mathtt{ISD-BC,}\ast}$ is given by \[ \tilde{\theta}_{n}^{\mathtt{ISD-BC,}\ast}=\hat{\theta}_{n}^{\mathtt{ISD-BC,} \ast}-\frac{\int_{\mathbb{R}^{d}}K(u)^{2}du}{nh_{n}^{d}}. \] In perfect analogy with $\tilde{\theta}_{n}^{\mathtt{AD-BC,}\ast},$ this estimator satisfies $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ and the associated percentile interval is identical to the percentile interval associated with $\hat{\theta}_{n}^{\mathtt{ISD,}\ast}.$ Next, \[ \tilde{\theta}_{n}^{\mathtt{ISD-LO,}\ast}=\frac{1}{n}\sum_{1\leq i\leq n} \int_{\mathbb{R}^{d}}\tilde{f}_{i,n}^{\mathtt{LO,}\ast}(x)^{2}dx \] is an integrated squared density counterpart of $\tilde{\theta}_{n} ^{\mathtt{AD-LO,}\ast}.$ Because $\tilde{\theta}_{n}^{\mathtt{ISD-LO,}\ast }=\hat{\theta}_{n}^{\mathtt{ISD-LO,}\ast},$ this estimator satisfies $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ when $B_{n}=n,$ but not when $B_{n}$ is fixed. On the other hand, the cross-fit bootstrap can be used when $B_{n}$ is fixed. As before, suppose $B_{n}=2$ for specificity. In that case, $\hat{\theta}_{n}^{\mathtt{ISD-LO}}$ reduces to \[ \hat{\theta}_{n}^{\mathtt{ISD-CF}}=\frac{1}{n}\sum_{1\leq i\leq n} \int_{\mathbb{R}^{d}}\hat{f}_{i,n}^{\mathtt{CF}}(x)^{2}dx \] and it can be shown that \[ \hat{\theta}_{n}^{\mathtt{ISD-CF,}\star}=\frac{1}{n}\sum_{1\leq i\leq n} \int_{\mathbb{R}^{d}}\hat{f}_{i,n}^{\mathtt{CF,}\star}(x)^{2}dx \] satisfies $\left( \ref{Bootstrap consistency: Alternative measure}\right) .$ Similarly, the distribution of $\hat{\theta}_{n}^{\mathtt{ISD-DCF}}$ can be approximated using \[ \hat{\theta}_{n}^{\mathtt{ISD-DCF,}\star}=\int_{\mathbb{R}^{d}}\hat{f} _{1,n}^{\mathtt{CF,}\star}\left( x\right) \hat{f}_{n,n}^{\mathtt{CF,}\star }\left( x\right) dx, \] as that estimator satisfies $\left( \ref{Bootstrap consistency: Alternative measure}\right) .$ Finally, the property $\left( \ref{Bootstrap consistency: Alternative centering}\right) $ holds for $\hat{\theta}_{n}^{\mathtt{ISD-BC}},$ $\hat{\theta}_{n} ^{\mathtt{ISD-GJ}},$ and $\hat{\theta}_{n}^{\mathtt{ISD-DCF}}$ when $\tilde{\theta}_{n}=\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\ast}]$ (and for $\hat{\theta}_{n}^{\mathtt{ISD}},$ $\hat{\theta}_{n}^{\mathtt{ISD-GJ}},$ and the leave-one-out version of $\hat{\theta}_{n}^{\mathtt{ISD-LO}}$ when $\tilde{\theta}_{n}=\hat{\theta}_{n}$).
A locally robust kernel-based plug-in\ estimator of $\theta_{0}$ is \[ \hat{\theta}_{n}^{\mathtt{LR}}=\frac{2}{n}\sum_{1\leq i\leq n}\hat{f} _{n}(X_{i})-\int_{\mathbb{R}^{d}}\hat{f}_{n}\left( x\right) ^{2} dx=2\hat{\theta}_{n}^{\mathtt{AD}}-\hat{\theta}_{n}^{\mathtt{ISD}}. \] Because $\hat{\theta}_{n}^{\mathtt{LR}}$ is a linear combination of $\hat{\theta}_{n}^{\mathtt{AD}}$ and $\hat{\theta}_{n}^{\mathtt{ISD}},$ its properties follow directly from the results obtained in the previous sections, as do the properties of estimators such as \[ \hat{\theta}_{n}^{\mathtt{LR-BC}}=2\hat{\theta}_{n}^{\mathtt{AD-BC}} -\hat{\theta}_{n}^{\mathtt{ISD-BC}}, \] \[ \hat{\theta}_{n}^{\mathtt{LR-GJ}}=2\hat{\theta}_{n}^{\mathtt{AD-GJ}} -\hat{\theta}_{n}^{\mathtt{ISD-GJ}}, \] and \[ \hat{\theta}_{n}^{\mathtt{LR-LO}}=2\hat{\theta}_{n}^{\mathtt{AD-LO}} -\hat{\theta}_{n}^{\mathtt{ISD-LO}}, \] the cross-fit version of the latter being the only estimator (in this paper) satisfying both of the defining properties of the \textquotedblleft double/debiased machine learning\textquotedblright\ estimators proposed by \citet*{Chernozhukov-Chetverikov-Demirer-Duflo-Hansen-Newey-Robins_2018_EctJ}.
Once again, the results are in qualitative agreement with those reported in Theorems (ref) and (ref).
Rather than spelling out those locally robust versions of $\left( \ref{Bootstrap consistency: Alternative estimator}\right) ,$ $\left( \ref{Bootstrap consistency: Alternative measure}\right) $, and $\left( \ref{Bootstrap consistency: Alternative centering}\right) $ that follow directly from our earlier results, it seems more constructive to mention a feature of local robustness that is particularly useful for boostrap purposes. As pointed out by \citet*{Belloni-Chernozhukov-FernandezVal-Hansen_2017_ECMA}, a notable feature of locally robust moment conditions is that in two-step estimation settings one does not need to recompute the first step estimator in each iteration of the bootstrap. In the case of $\hat{\theta}_{n} ^{\mathtt{LR}},$ this implies that $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ can be achieved with the help of \[ \tilde{\theta}_{n}^{\mathtt{LR,}\ast}=\frac{2}{n}\sum_{1\leq i\leq n}\hat {f}_{n}(X_{i,n}^{\ast})-\int_{\mathbb{R}^{d}}\hat{f}_{n}\left( x\right) ^{2}dx, \] a computationally attractive feature of which is that $\hat{f}_{n}$ is kept fixed across bootstrap repetitions. Perhaps more importantly (for our purposes at least), the fact that $\hat{f}_{n}$ is kept fixed actually makes it easier to achieve $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ also in the case of debiased estimators. For instance, \[ \tilde{\theta}_{n}^{\mathtt{LR-BC,}\ast}=\frac{2}{n}\sum_{1\leq i\leq n} \hat{f}_{n}(X_{i,n}^{\ast})-\int_{\mathbb{R}^{d}}\hat{f}_{n}\left( x\right) ^{2}dx-\frac{2K(0)-\int_{\mathbb{R}^{d}}K(u)^{2}du}{nh_{n}^{d}} \] satisfies $\left( \ref{Bootstrap consistency: Alternative estimator}\right) $ under Conditions D, K, and B.
Each of the estimators studied in this paper has a $V$-statistic-type representation of the form \[ \hat{\theta}_{n}=\frac{1}{n^{2}}\sum_{1\leq i,j\leq n}V_{ij,n}, \] where $V_{ij,n}$ depends on $X_{1},\ldots,X_{n}$ only through $(X_{i},X_{j}).$ The proofs of Theorems (ref), (ref), and (ref) are based on the associated Hoeffding decomposition of $\hat{\theta}_{n} -\theta_{0}$ given by
where, defining $\bar{V}_{ij,n}=(V_{ij,n}+V_{ji,n})/2,$
By construction, $L_{i,n}$ and $W_{ij,n}$ depend on $X_{1},\ldots,X_{n}$ only through $X_{i}$ and $(X_{i},X_{j}),$ respectively, and satisfy, for each $1\leq i,j\leq n$ with $i\neq j,$ \[ \mathbb{E}[L_{i,n}]=\mathbb{E}[W_{ij,n}|X_{i}]=\mathbb{E}[W_{ij,n}|X_{j}]=0. \] Moreover, if the $V_{ij,n}$ satisfy $V_{ii,n}=\delta_{n}$ and $\mathbb{E} [V_{ij,n}]=\theta_{n},$ then the bias is of the form \[ \beta_{n}=\frac{\delta_{n}}{n}+\theta_{n}-\theta_{0}-\frac{\theta_{n}}{n}. \] If also $V_{ij,n}=V_{ji,n}$ and $\mathbb{E}[V_{ij,n}|X_{i}]=f_{n}(X_{i}),$ then \[ L_{i,n}=2\frac{n-1}{n}\{f_{n}(X_{i})-\theta_{n}\},\qquad W_{ij,n}=\frac {n-1}{n}\{V_{ij,n}-f_{n}(X_{i})-f_{n}(X_{j})+\theta_{n}\}. \]
A bootstrap analog of $\left( \ref{Hoeffding decomposition}\right) $ will be employed in the proofs of Theorems (ref), (ref), and (ref). To state it, suppose \[ \hat{\theta}_{n}^{\ast}=\frac{1}{n^{2}}\sum_{1\leq i,j\leq n}V_{ij,n}^{\ast}, \] where $V_{ij,n}^{\ast}$ depends on $X_{1,n}^{\ast},\ldots,X_{n,n}^{\ast}$ only through $(X_{i,n}^{\ast},X_{j,n}^{\ast}).$ Then
where, defining $\bar{V}_{ij,n}^{\ast}=(V_{ij,n}^{\ast}+V_{ji,n}^{\ast})/2,$
By construction, $L_{i,n}^{\ast}$ and $W_{ij,n}^{\ast}$ depend on $X_{1,n}^{\ast},\ldots,X_{n,n}^{\ast}$ only through $X_{i,n}^{\ast}$ and $(X_{i,n}^{\ast},X_{j,n}^{\ast}),$ respectively, and satisfy, for each $1\leq i,j\leq n$ with $i\neq j,$ \[ \mathbb{E}_{n}^{\ast}[L_{i,n}^{\ast}]=\mathbb{E}_{n}^{\ast}[W_{ij,n}^{\ast }|X_{i,n}^{\ast}]=\mathbb{E}_{n}^{\ast}[W_{ij,n}^{\ast}|X_{j,n}^{\ast}]=0. \] Moreover, if the $V_{ij,n}^{\ast}$ satisfy $V_{ii,n}^{\ast}=\delta_{n}^{\ast}$ and $\mathbb{E}_{n}^{\ast}[V_{ij,n}^{\ast}]=\theta_{n}^{\ast},$ then the bootstrap bias is of the form \[ \beta_{n}^{\ast}=\frac{\delta_{n}^{\ast}}{n}+\theta_{n}^{\ast}-\hat{\theta }_{n}-\frac{\theta_{n}^{\ast}}{n}. \] If also $V_{ij,n}^{\ast}=V_{ji,n}^{\ast}$ and $\mathbb{E}_{n}^{\ast} [V_{ij,n}^{\ast}|X_{i,n}^{\ast}]=f_{n}^{\ast}(X_{i,n}^{\ast}),$ then \[ L_{i,n}^{\ast}=2\frac{n-1}{n}\{f_{n}^{\ast}(X_{i,n}^{\ast})-\theta_{n}^{\ast }\},\qquad W_{ij,n}^{\ast}=\frac{n-1}{n}\{V_{ij,n}^{\ast}-f_{n}^{\ast} (X_{i,n}^{\ast})-f_{n}^{\ast}(X_{j,n}^{\ast})+\theta_{n}^{\ast}\}. \]
The estimators $\hat{\theta}_{n}^{\mathtt{AD}}$ and $\hat{\theta} _{n}^{\mathtt{AD-LO}}$ both have Hoeffding decompositions of the form $\left( \ref{Hoeffding decomposition}\right) ,$ with \[ L_{i,n}=\lambda_{i,n}L_{n}^{\mathtt{AD}}(X_{i})\text{\qquad and\qquad} W_{ij,n}=\omega_{ij,n}W_{n}^{\mathtt{AD}}(X_{i},X_{j}), \] where $\lambda_{i,n}$ and $\omega_{ij,n}$ are (non-random) estimator-specific weights, while \[ L_{n}^{\mathtt{AD}}(x)=2\{f_{n}^{\mathtt{AD}}(x)-\theta_{n}^{\mathtt{AD}}\}, \] \[ W_{n}^{\mathtt{AD}}(x_{1},x_{2})=K_{n}(x_{1}-x_{2})-f_{n}^{\mathtt{AD}} (x_{1})-f_{n}^{\mathtt{AD}}(x_{2})+\theta_{n}^{\mathtt{AD}}, \] where
\[ f_{n}^{\mathtt{AD}}\left( x\right) =\mathbb{E}[K_{n}(x-X)]=\int _{\mathbb{R}^{d}}K(u)f_{0}(x+uh_{n})du, \] \[ \theta_{n}^{\mathtt{AD}}=\mathbb{E}[f_{n}^{\mathtt{AD}}(X)]=\int _{\mathbb{R}^{d}}f_{n}^{\mathtt{AD}}(x)f_{0}(x)dx. \]
To be specific, in the case of \[ \hat{\theta}_{n}^{\mathtt{AD}}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{n}(X_{i})=\frac{1}{n^{2}}\sum_{1\leq i,j\leq n}K_{n}(X_{i}-X_{j}), \] each $\lambda_{i,n}$ and $\omega_{ij,n}$ is given by $1-n^{-1},$ while the weights for \[ \hat{\theta}_{n}^{\mathtt{AD-LO}}=\frac{1}{n}\sum_{1\leq i\leq n}\hat{f} _{i,n}^{\mathtt{LO}}(X_{i})=\frac{1}{n^{2}}\sum_{1\leq i,j\leq n} nw_{ij,n}K_{n}(X_{i}-X_{j}) \] are of the form \[ \lambda_{i,n}=\sum_{1\leq j\leq n}\bar{w}_{ij,n},\qquad\omega_{ij,n} =(n-1)\bar{w}_{ij,n},\qquad\bar{w}_{ij,n}=(w_{ij,n}+w_{ji,n})/2. \]
In both cases, the weights satisfy
and
It therefore follows from simple moment calculations that the estimators satisfy $\left( \ref{Efficiency: Noise condition}\right) $ if
and if
Suppose Conditions D and K are satisfied. Then $\left( \ref{Quadratic term: variance}\right) $ holds if $nh_{n}^{d}\rightarrow \infty,$ because then
Also, because \[ \mathbb{E}[\{L_{n}^{\mathtt{AD}}(X)-L_{0}(X)\}^{2}]\leq4\mathbb{E}\left[ \{f_{n}^{\mathtt{AD}}(X)-f_{0}(X)\}^{2}\right] , \] a sufficient condition for $\left( \ref{Linear term: mean square convergence} \right) $ to hold is that \[ \mathbb{E}\left[ \{f_{n}^{\mathtt{AD}}(X)-f_{0}(X)\}^{2}\right] \rightarrow0. \] As in Proposition 1(c) of \citet*{Gine-Nickl_2008_PTRF}, the displayed condition is satisfied if $h_{n}\rightarrow0.$ To summarize, each estimator satisfies $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$.
The proof will be completed by giving conditions under which the estimators satisfy $\left( \ref{Efficiency: Bias condition}\right) .$ As before, suppose Conditions D and K are satisfied. In the notation introduced above, the biases of $\hat{\theta}_{n}^{\mathtt{AD}}$ and $\hat{\theta} _{n}^{\mathtt{AD-LO}}$ are given by \[ \beta_{n}^{\mathtt{AD}}=\frac{K(0)}{nh_{n}^{d}}+\theta_{n}^{\mathtt{AD} }-\theta_{0}-\frac{\theta_{n}^{\mathtt{AD}}}{n} \] and \[ \beta_{n}^{\mathtt{AD-LO}}=\theta_{n}^{\mathtt{AD}}-\theta_{0}, \] respectively. Following \citet*{Gine-Nickl_2008_Bernoulli}, we base our analysis of the smoothing bias $\theta_{n}^{\mathtt{AD}}-\theta_{0}$ on the representation
where the last equality uses the fact that $K$ is even. By Lemma 12 of \citet*{Gine-Nickl_2008_PTRF}, the function $f_{0}^{\Delta}$ belongs to the H\"{o}lder space $\mathbf{C}^{2s}(\mathbb{R}^{d}).$ As a consequence, it follows from standard arguments (e.g., \citet*[Proposition 1.2] {Tsybakov_2009_Book}) that if Condition B is satisfied, then \[ \theta_{n}^{\mathtt{AD}}-\theta_{0}=\int_{\mathbb{R}^{d}}K(t)[f_{0}^{\Delta }(h_{n}t)-f_{0}^{\Delta}(0)]dt=O(h_{n}^{S})=o(n^{-1/2}). \]
In particular, $\hat{\theta}_{n}^{\mathtt{AD-LO}}$ satisfies $\left( \ref{Efficiency: Bias condition}\right) $ under Conditions D, K, and B. Under the same conditions, $\theta_{n}^{\mathtt{AD}}$ is bounded, so \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{AD}}]-\theta_{0})=\frac {K(0)}{\sqrt{nh_{n}^{2d}}}+o(1), \] implying in particular that Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{AD}}$ to satisfy $\left( \ref{Efficiency: Bias condition}\right) $ (unless $K(0)=0$).
Finally, the results for $\hat{\theta}_{n}^{\mathtt{AD-BC}}$ and $\hat{\theta }_{n}^{\mathtt{AD-GJ}}$ follow from those for $\hat{\theta}_{n}^{\mathtt{AD} }.$ To be specific, $\hat{\theta}_{n}^{\mathtt{AD-BC}}$ differs from $\hat{\theta}_{n}^{\mathtt{AD}}$ by an additive constant, so it satisfies $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$. Also, the additive constant is designed to ensure that $\left( \ref{Efficiency: Bias condition}\right) $ is satisfied by $\hat{\theta} _{n}^{\mathtt{AD-BC}}$ under Conditions D, K, and B. Similarly, because \[ \frac{1}{1-c^{d}}-\frac{c^{d}}{1-c^{d}}=1, \] the estimator $\hat{\theta}_{n}^{\mathtt{AD-GJ}}$ satisfies $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D, K, and B$^{-} $, while the fact that \[ \frac{1}{1-c^{d}}\frac{1}{nh_{n}^{d}}-\frac{c^{d}}{1-c^{d}}\frac{1} {n(ch_{n})^{d}}=0 \] ensures that $\left( \ref{Efficiency: Bias condition}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{AD-GJ}}$ under Conditions D, K, and B.
The estimators $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ and $\hat{\theta} _{n}^{\mathtt{AD-LO,}\ast}$ both have Hoeffding decompositions of the form $\left( \ref{Hoeffding decomposition (bootstrap)}\right) ,$ with \[ L_{i,n}^{\ast}=\lambda_{i,n}\hat{L}_{n}^{\mathtt{AD}}(X_{i,n}^{\ast })\text{\qquad and\qquad}W_{ij,n}^{\ast}=\omega_{ij,n}\hat{W}_{n} ^{\mathtt{AD}}(X_{i,n}^{\ast},X_{j,n}^{\ast}), \] where $\lambda_{i,n}$ and $\omega_{ij,n}$ are the same as those for $\hat{\theta}_{n}^{\mathtt{AD}}$ and $\hat{\theta}_{n}^{\mathtt{AD-LO}},$ while \[ \hat{L}_{n}^{\mathtt{AD}}(x)=2\{\hat{f}_{n}(x)-\hat{\theta}_{n}^{\mathtt{AD} }\}, \] \[ \hat{W}_{n}^{\mathtt{AD}}(x_{1},x_{2})=K_{n}(x_{1}-x_{2})-\hat{f}_{n} (x_{1})-\hat{f}_{n}(x_{2})+\hat{\theta}_{n}^{\mathtt{AD}}. \]
Because the weights satisfy $\left( \ref{Linear term: weights}\right) $ and $\left( \ref{Quadratic term: weights}\right) ,$ it follows from simple moment calculations that the estimators satisfy \[ \sqrt{n}(\hat{\theta}_{n}^{\ast}-\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\ast }])=\frac{1}{\sqrt{n}}\sum_{1\leq i\leq n}\{L_{0}(X_{i,n}^{\ast} )-\mathbb{E}_{n}^{\ast}[L_{0}(X_{i,n}^{\ast})]\}+o_{\mathbb{P}} (1)\rightsquigarrow_{\mathbb{P}}\mathcal{N}(0,\sigma_{0}^{2}) \] if
and if $\left( \ref{Linear term: mean square convergence}\right) $ and $\left( \ref{Linear term (bootstrap): mean square convergence}\right) $ hold, where
Suppose Conditions D and K are satisfied. Then $\left( \ref{Quadratic term (bootstrap): variance}\right) $ holds if $nh_{n} ^{d}\rightarrow\infty,$ because then
where the convergence result follow from the proof of Theorem (ref). In that same proof it was shown that $\left( \ref{Linear term: mean square convergence}\right) $ holds when $h_{n}\rightarrow0.$ Finally, because \[ \mathbb{E}_{n}^{\ast}[\{\hat{L}_{n}^{\mathtt{AD}}(X_{1,n}^{\ast} )-L_{n}^{\mathtt{AD}}(X_{1,n}^{\ast})\}^{2}]=\frac{1}{n}\sum_{1\leq i\leq n}\{\hat{L}_{n}^{\mathtt{AD}}(X_{i})-L_{n}^{\mathtt{AD}}(X_{i})\}^{2}, \] a sufficient condition for $\left( \ref{Linear term (bootstrap): mean square convergence}\right) $ to hold is that \[ \mathbb{E}[\{\hat{L}_{n}^{\mathtt{AD}}(X_{1})-L_{n}^{\mathtt{AD}}(X_{1} )\}^{2}]\rightarrow0. \] It follows from a direct calculation this condition is satisfied when $h_{n}\rightarrow0$ and $nh_{n}^{d}\rightarrow\infty.$ To summarize, each estimator satisfies $\left( \ref{Bootstrap consistency: Noise condition} \right) $ under Conditions D, K, and B$^{-}$.
The proof will be completed by giving conditions under which the estimators satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) .$ Suppose Conditions D, K, and B are satisfied. By the proof of Theorem (ref), \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{AD}}]-\theta_{0})=\frac {K(0)}{\sqrt{nh_{n}^{2d}}}+o(1), \] and \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{AD-LO}}]-\theta_{0})=o(1), \] while it follows from $\left( \ref{Hoeffding decomposition (bootstrap)} \right) $ and Theorem (ref) that \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{AD,}\ast} ]-\hat{\theta}_{n}^{\mathtt{AD}})=\frac{K(0)}{\sqrt{nh_{n}^{2d}}}-\frac {\hat{\theta}_{n}^{\mathtt{AD}}}{\sqrt{n}}=\frac{K(0)}{\sqrt{nh_{n}^{2d}} }+o_{\mathbb{P}}(1), \] and \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{AD-LO,}\ast} ]-\hat{\theta}_{n}^{\mathtt{AD-LO}})=\sqrt{n}(\hat{\theta}_{n}^{\mathtt{AD} }-\hat{\theta}_{n}^{\mathtt{AD-LO}})=\frac{K(0)}{\sqrt{nh_{n}^{2d}} }+o_{\mathbb{P}}(1). \] As a consequence, $\hat{\theta}_{n}^{\mathtt{AD,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Bias condition}\right) $ under Conditions D, K, and B, whereas Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{AD-LO,}\ast}$ to satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) $ (unless $K(0)=0$).
Finally, the results for $\hat{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ and $\hat{\theta}_{n}^{\mathtt{AD-GJ,}\ast}$ follow from those for $\hat{\theta }_{n}^{\mathtt{AD,}\ast}.$ To be specific, $\hat{\theta}_{n}^{\mathtt{AD-BC,} \ast} $ satisfies $\left( \ref{Bootstrap consistency: Noise condition} \right) $ under Conditions D, K, and B$^{-}$ because $\hat{\theta} _{n}^{\mathtt{AD,}\ast}$ does. Moreover, \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{AD-BC,}\ast} ]-\hat{\theta}_{n}^{\mathtt{AD-BC}})=\sqrt{n}(\mathbb{E}_{n}^{\ast} [\hat{\theta}_{n}^{\mathtt{AD,}\ast}]-\hat{\theta}_{n}^{\mathtt{AD}}), \] so under Conditions D and K, Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{AD-BC,}\ast}$ to satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) $ (unless $K(0)=0$). Similarly, because \[ \frac{1}{1-c^{d}}-\frac{c^{d}}{1-c^{d}}=1, \] the estimator $\hat{\theta}_{n}^{\mathtt{AD-GJ,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$, while the fact that \[ \frac{1}{1-c^{d}}\frac{1}{nh_{n}^{d}}-\frac{c^{d}}{1-c^{d}}\frac{1} {n(ch_{n})^{d}}=0 \] ensures that $\left( \ref{Bootstrap consistency: Bias condition}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{AD-GJ,}\ast}$ under Conditions D, K, and B.
The proof is similar to that of Theorem (ref). The estimators $\hat{\theta}_{n}^{\mathtt{ISD}},\ \hat{\theta}_{n} ^{\mathtt{ISD-LO}},$ and $\hat{\theta}_{n}^{\mathtt{ISD-CF}}$ all have Hoeffding decompositions of the form $\left( \ref{Hoeffding decomposition} \right) ,$ with \[ L_{i,n}=\lambda_{i,n}L_{n}^{\mathtt{ISD}}(X_{i}),\text{\qquad}W_{ij,n} =\omega_{ij,n}W_{n}^{\mathtt{ISD}}(X_{i},X_{j}), \] where $\lambda_{i,n}$ and $\omega_{ij,n}$ are (non-random) estimator-specific weights, while \[ L_{n}^{\mathtt{ISD}}(x)=2\{f_{n}^{\mathtt{ISD}}(x)-\theta_{n}^{\mathtt{ISD} }\}, \] \[ W_{n}^{\mathtt{ISD}}(x_{1},x_{2})=K_{n}^{\Delta}(x_{1}-x_{2})-f_{n} ^{\mathtt{ISD}}(x_{1})-f_{n}^{\mathtt{ISD}}(x_{2})+\theta_{n}^{\mathtt{ISD}}, \] where
\[ f_{n}^{\mathtt{ISD}}\left( x\right) =\mathbb{E}[K_{n}^{\Delta} (x-X)]=\int_{\mathbb{R}^{d}}K^{\Delta}(u)f_{0}(x+uh_{n})du, \] \[ \theta_{n}^{\mathtt{ISD}}=\mathbb{E}[f_{n}^{\mathtt{ISD}}(X)]=\int _{\mathbb{R}^{d}}f_{n}^{\mathtt{ISD}}(x)f_{0}(x)dx, \] \[ K_{n}^{\Delta}(x)=\frac{1}{h_{n}^{d}}K^{\Delta}\left( \frac{x}{h_{n}}\right) ,\qquad K^{\Delta}(x)=\int_{\mathbb{R}^{d}}K(u)K(x+u)du. \]
To be specific, in the case of
each $\lambda_{i,n}$ and $\omega_{ij,n}$ is given by $1-n^{-1}.$ For
the weights are given by \[ \lambda_{i,n}=\sum_{1\leq j,k\leq n,j\neq i}w_{ki,n}w_{kj,n},\qquad \omega_{ij,n}=(n-1)\sum_{1\leq k\leq n}w_{ki,n}w_{kj,n}, \] while the weights for
can be shown to be given by
\[ \lambda_{i,n}=\frac{n/2}{\sum_{1\leq j\leq n} {\rm 1\hspace*{-0.4ex}\rule{0.1ex}{1.52ex}\hspace*{0.2ex}} (\left\lceil 2i/n\right\rceil =\left\lceil 2j/n\right\rceil )},\qquad \omega_{ij,n}^{\mathtt{ISD-DCF}}=\frac{n(n-1)/2}{(n-\left\lfloor n/2\right\rfloor )\left\lfloor n/2\right\rfloor } {\rm 1\hspace*{-0.4ex}\rule{0.1ex}{1.52ex}\hspace*{0.2ex}} (\left\lceil 2i/n\right\rceil \neq\left\lceil 2j/n\right\rceil ). \]
In all cases, the weights satisfy $\left( \ref{Linear term: weights}\right) $ and $\left( \ref{Quadratic term: weights}\right) ,$ so the estimators satisfy $\left( \ref{Efficiency: Noise condition}\right) $ if
and if
Proceeding as in the proof of Theorem (ref) it can be shown that $\left( \ref{Quadratic term: variance (ISD)}\right) $ and $\left( \ref{Linear term: mean square convergence (ISD)}\right) $ hold under Conditions D, K, and B$^{-}$.
Finally, the biases of $\hat{\theta}_{n}^{\mathtt{ISD}},\ \hat{\theta} _{n}^{\mathtt{ISD-LO}},$ and $\hat{\theta}_{n}^{\mathtt{ISD-DCF}}$ are given by \[ \beta_{n}^{\mathtt{ISD}}=\frac{K^{\Delta}(0)}{nh_{n}^{d}}+\theta _{n}^{\mathtt{ISD}}-\theta_{0}-\frac{\theta_{n}^{\mathtt{ISD}}}{n}, \] \[ \beta_{n}^{\mathtt{ISD-LO}}=\eta_{n}\frac{K^{\Delta}(0)}{nh_{n}^{d}} +\theta_{n}^{\mathtt{ISD}}-\theta_{0}-\eta_{n}\frac{\theta_{n}^{\mathtt{ISD}} }{n}, \] and \[ \beta_{n}^{\mathtt{ISD-DCF}}=\theta_{n}^{\mathtt{ISD}}-\theta_{0}, \] respectively, where \[ \eta_{n}=\sum_{1\leq i\leq n}\frac{1}{\sum_{1\leq j\leq n} {\rm 1\hspace*{-0.4ex}\rule{0.1ex}{1.52ex}\hspace*{0.2ex}} (\left\lceil iB_{n}/n\right\rceil \neq\left\lceil jB_{n}/n\right\rceil )}, \] and where \[ \theta_{n}^{\mathtt{ISD}}-\theta_{0}=\int_{\mathbb{R}^{d}}K^{\Delta} (t)[f_{0}^{\Delta}(h_{n}t)-f_{0}^{\Delta}(0)]dt=O(h_{n}^{S})=o(n^{-1/2}) \] under Conditions D, K, and B.
As a consequence $\hat{\theta}_{n}^{\mathtt{ISD-DCF}}$ satisfies $\left( \ref{Efficiency: Bias condition}\right) $ under Conditions D, K, and B, whereas \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{ISD}}]-\theta_{0})=\frac {K^{\Delta}(0)}{\sqrt{nh_{n}^{2d}}}+o(1), \] so Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta} _{n}^{\mathtt{ISD}}$ to satisfy $\left( \ref{Efficiency: Bias condition} \right) .$ Finally, $\eta_{n}\geq1$ is bounded, so Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{ISD-LO}}$ to satisfy $\left( \ref{Efficiency: Bias condition}\right) .$
Finally, the results for $\hat{\theta}_{n}^{\mathtt{ISD-BC}}$ and $\hat {\theta}_{n}^{\mathtt{ISD-GJ}}$ follow from those for $\hat{\theta} _{n}^{\mathtt{ISD}}.$ To be specific, $\hat{\theta}_{n}^{\mathtt{ISD-BC}}$ differs from $\hat{\theta}_{n}^{\mathtt{ISD}}$ by an additive constant, so it satisfies $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$. Also, the additive constant is designed to ensure that $\left( \ref{Efficiency: Bias condition}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{ISD-BC}}$ under Conditions D, K, and B. Similarly, because \[ \frac{1}{1-c^{d}}-\frac{c^{d}}{1-c^{d}}=1, \] the estimator $\hat{\theta}_{n}^{\mathtt{ISD-GJ}}$ satisfies $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D, K, and B$^{-} $, while the fact that \[ \frac{1}{1-c^{d}}\frac{1}{nh_{n}^{d}}-\frac{c^{d}}{1-c^{d}}\frac{1} {n(ch_{n})^{d}}=0 \] ensures that $\left( \ref{Efficiency: Bias condition}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{ISD-GJ}}$ under Conditions D, K, and B.
The proof is similar to that of Theorem (ref). The estimators $\hat{\theta}_{n}^{\mathtt{ISD,}\ast},\ \hat{\theta} _{n}^{\mathtt{ISD-LO,}\ast},$ and $\hat{\theta}_{n}^{\mathtt{ISD-DCF,}\ast}$ all have Hoeffding decompositions of the form $\left( \ref{Hoeffding decomposition (bootstrap)}\right) ,$ with \[ L_{i,n}^{\ast}=\lambda_{i,n}\hat{L}_{n}^{\mathtt{ISD}}(X_{i,n}^{\ast }),\text{\qquad}W_{ij,n}^{\ast}=\omega_{ij,n}\hat{W}_{n}^{\mathtt{ISD} }(X_{i,n}^{\ast},X_{j,n}^{\ast}), \] where $\lambda_{i,n}$ and $\omega_{ij,n}$ are the same as those for $\hat{\theta}_{n}^{\mathtt{ISD}},\ \hat{\theta}_{n}^{\mathtt{ISD-LO}},$ and $\hat{\theta}_{n}^{\mathtt{ISD-DCF}},$ while \[ \hat{L}_{n}^{\mathtt{ISD}}(x)=2\{\hat{f}_{n}^{\mathtt{ISD}}(x)-\hat{\theta }_{n}^{\mathtt{ISD}}\},\qquad\hat{f}_{n}^{\mathtt{ISD}}(x)=\frac{1}{n} \sum_{1\leq j\leq n}K_{n}^{\Delta}(x-X_{j}), \] \[ \hat{W}_{n}^{\mathtt{ISD}}(x_{1},x_{2})=K_{n}^{\Delta}(x_{1}-x_{2})-\hat {f}_{n}^{\mathtt{ISD}}(x_{1})-\hat{f}_{n}^{\mathtt{ISD}}(x_{2})+\hat{\theta }_{n}^{\mathtt{ISD}}. \]
Because the weights satisfy $\left( \ref{Linear term: weights}\right) $ and $\left( \ref{Quadratic term: weights}\right) ,$ it follows from simple moment calculations that the estimators satisfy \[ \sqrt{n}(\hat{\theta}_{n}^{\ast}-\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\ast }])=\frac{1}{\sqrt{n}}\sum_{1\leq i\leq n}\{L_{0}(X_{i,n}^{\ast} )-\mathbb{E}_{n}^{\ast}[L_{0}(X_{i,n}^{\ast})]\}+o_{\mathbb{P}} (1)\rightsquigarrow_{\mathbb{P}}\mathcal{N}(0,\sigma_{0}^{2}) \] if
and if $\left( \ref{Linear term: mean square convergence (ISD)}\right) $ and $\left( \ref{Linear term (bootstrap): mean square convergence (ISD)}\right) $ hold, where
Suppose Conditions D and K are satisfied. Then $\left( \ref{Quadratic term (bootstrap): variance (ISD)}\right) $ holds if $nh_{n}^{d}\rightarrow\infty,$ because then
Also, $\left( \ref{Linear term: mean square convergence (ISD)}\right) $ holds when $h_{n}\rightarrow0.$ Finally, because \[ \mathbb{E}_{n}^{\ast}[\{\hat{L}_{n}^{\mathtt{ISD}}(X_{1,n}^{\ast} )-L_{n}^{\mathtt{ISD}}(X_{1,n}^{\ast})\}^{2}]=\frac{1}{n}\sum_{1\leq i\leq n}\{\hat{L}_{n}^{\mathtt{ISD}}(X_{i})-L_{n}^{\mathtt{ISD}}(X_{i})\}^{2}, \] a sufficient condition for $\left( \ref{Linear term (bootstrap): mean square convergence (ISD)}\right) $ to hold is that \[ \mathbb{E}[\{\hat{L}_{n}^{\mathtt{ISD}}(X_{1})-L_{n}^{\mathtt{ISD}} (X_{1})\}^{2}]\rightarrow0. \] It follows from a direct calculation this condition is satisfied when $h_{n}\rightarrow0$ and $nh_{n}^{d}\rightarrow\infty.$ To summarize, each estimator satisfies $\left( \ref{Bootstrap consistency: Noise condition} \right) $ under Conditions D, K, and B$^{-}$.
The proof will be completed by giving conditions under which the estimators satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) .$ Suppose Conditions D, K, and B are satisfied. By the proof of Theorem (ref), \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{ISD}}]-\theta_{0})=\frac {K^{\Delta}(0)}{\sqrt{nh_{n}^{2d}}}+o(1), \] \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{ISD-LO}}]-\theta_{0})=\eta _{n}\frac{K^{\Delta}(0)}{\sqrt{nh_{n}^{2d}}}+o(1), \] and \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{ISD-DCF}}]-\theta_{0})=o(1), \] while it follows from $\left( \ref{Hoeffding decomposition (bootstrap)} \right) $ and Theorem (ref) that \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{ISD,}\ast} ]-\hat{\theta}_{n}^{\mathtt{ISD}})=\frac{K^{\Delta}(0)}{\sqrt{nh_{n}^{2d}} }-\frac{\hat{\theta}_{n}^{\mathtt{ISD}}}{\sqrt{n}}=\frac{K^{\Delta}(0)} {\sqrt{nh_{n}^{2d}}}+o_{\mathbb{P}}(1), \] \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{ISD-LO,}\ast} ]-\hat{\theta}_{n}^{\mathtt{ISD-LO}})=\eta_{n}\frac{K^{\Delta}(0)} {\sqrt{nh_{n}^{2d}}}+\sqrt{n}(\hat{\theta}_{n}^{\mathtt{ISD}}-\hat{\theta} _{n}^{\mathtt{ISD-LO}})-\eta_{n}\frac{\hat{\theta}_{n}^{\mathtt{ISD}}} {\sqrt{n}}=\frac{K^{\Delta}(0)}{\sqrt{nh_{n}^{2d}}}+o_{\mathbb{P}}(1). \] and \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{ISD-DCF,}\ast} ]-\hat{\theta}_{n}^{\mathtt{ISD-DCF}})=\sqrt{n}(\hat{\theta}_{n} ^{\mathtt{ISD}}-\hat{\theta}_{n}^{\mathtt{ISD-DCF}})=\frac{K^{\Delta} (0)}{\sqrt{nh_{n}^{2d}}}+o_{\mathbb{P}}(1). \] As a consequence, $\hat{\theta}_{n}^{\mathtt{ISD,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Bias condition}\right) $ under Conditions D, K, and B, whereas Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{ISD-DCF,}\ast}$ to satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) .$ Finally, if $B_{n}=n,$ then \[ \eta_{n}=\frac{n}{n-1}=1+O(n^{-1}), \] so $\hat{\theta}_{n}^{\mathtt{ISD-LO,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Bias condition}\right) $ under Conditions D, K, and B. Under the other hand, Condition B must be strengthened to Condition B$^{+} $ for the cross-fit version of $\hat{\theta}_{n}^{\mathtt{ISD-LO,}\ast} $ to satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) $ because if $B_{n}=B$ for all $n,$ then \[ \eta_{n}\rightarrow\frac{B}{B-1}\neq1. \]
Finally, the results for $\hat{\theta}_{n}^{\mathtt{ISD-BC,}\ast}$ and $\hat{\theta}_{n}^{\mathtt{ISD-GJ,}\ast}$ follow from those for $\hat{\theta }_{n}^{\mathtt{ISD,}\ast}.$ To be specific, $\hat{\theta}_{n} ^{\mathtt{ISD-BC,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$ because $\hat{\theta}_{n}^{\mathtt{ISD,}\ast}$ does. Moreover, \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{ISD-BC,}\ast} ]-\hat{\theta}_{n}^{\mathtt{ISD-BC}})=\sqrt{n}(\mathbb{E}_{n}^{\ast} [\hat{\theta}_{n}^{\mathtt{ISD,}\ast}]-\hat{\theta}_{n}^{\mathtt{ISD}}), \] so under Conditions D and K, Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{ISD-BC,}\ast}$ to satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) $ (unless $K(0)=0$). Similarly, because \[ \frac{1}{1-c^{d}}-\frac{c^{d}}{1-c^{d}}=1, \] the estimator $\hat{\theta}_{n}^{\mathtt{ISD-GJ,}\ast}$ satisfies $\left( \ref{Bootstrap consistency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$, while the fact that \[ \frac{1}{1-c^{d}}\frac{1}{nh_{n}^{d}}-\frac{c^{d}}{1-c^{d}}\frac{1} {n(ch_{n})^{d}}=0 \] ensures that $\left( \ref{Bootstrap consistency: Bias condition}\right) $ is satisfied by $\hat{\theta}_{n}^{\mathtt{ISD-GJ,}\ast}$ under Conditions D, K, and B.
It follows from the proofs of Theorems (ref) and (ref) that the estimators $\hat{\theta} _{n}^{\mathtt{LR}},$ $\hat{\theta}_{n}^{\mathtt{LR-BC}},$ $\hat{\theta} _{n}^{\mathtt{LR-GJ}},$ and $\hat{\theta}_{n}^{\mathtt{LR-LO}}$ satisfy $\left( \ref{Efficiency: Noise condition}\right) $ under Conditions D, K, and B$^{-}$ and have biases of the form \[ \beta_{n}^{\mathtt{LR}}=2\beta_{n}^{\mathtt{AD}}-\beta_{n}^{\mathtt{ISD} }=\frac{2K(0)-K^{\Delta}(0)}{nh_{n}^{d}}+o(n^{-1/2}), \] \[ \beta_{n}^{\mathtt{LR-BC}}=o(n^{-1/2}),\qquad\beta_{n}^{\mathtt{LR-GJ} }=o(n^{-1/2}), \] and \[ \beta_{n}^{\mathtt{LR-LO}}=2\beta_{n}^{\mathtt{AD-LO}}-\beta_{n} ^{\mathtt{ISD-LO}}=-\eta_{n}\frac{K^{\Delta}(0)}{nh_{n}^{d}}+o(n^{-1/2}), \] respectively, under Conditions D, K, and B.
As a consequence $\hat{\theta}_{n}^{\mathtt{LR-BC}}$ and $\hat{\theta} _{n}^{\mathtt{LR-GJ}}$ satisfy $\left( \ref{Efficiency: Bias condition} \right) $ under Conditions D, K, and B, whereas Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{LR-LO}}$ to satisfy $\left( \ref{Efficiency: Bias condition}\right) .$ Likewise, Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta} _{n}^{\mathtt{LR}}$ to satisfy $\left( \ref{Efficiency: Bias condition} \right) $ unless $2K(0)=K^{\Delta}(0).$
It follows from the proofs of Theorems (ref) and (ref) that the estimators $\hat{\theta}_{n}^{\mathtt{LR,}\ast},$ $\hat{\theta}_{n}^{\mathtt{LR-BC,}\ast },$ $\hat{\theta}_{n}^{\mathtt{LR-GJ,}\ast},$ and $\hat{\theta}_{n} ^{\mathtt{LR-LO,}\ast}$ satisfy $\left( \ref{Bootstrap consistency: Noise condition}\right) $ under Conditions D, K, and B$^{-}.$
The proof will be completed by giving conditions under which the estimators satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) .$ Suppose Conditions D, K, and B are satisfied. By the proof of Theorem (ref), \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{LR}}]-\theta_{0})=\frac {2K(0)-K^{\Delta}(0)}{\sqrt{nh_{n}^{2d}}}+o(1), \] \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{LR-BC}}]-\theta_{0})=o(1), \] \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{LR-GJ}}]-\theta_{0})=o(1), \] and \[ \sqrt{n}(\mathbb{E}[\hat{\theta}_{n}^{\mathtt{LR-LO}}]-\theta_{0})=-\eta _{n}\frac{K^{\Delta}(0)}{nh_{n}^{d}}+o(1), \] while it follows from the proofs of Theorems (ref) and (ref) that \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{LR,}\ast} ]-\hat{\theta}_{n}^{\mathtt{LR}})=\frac{2K(0)-K^{\Delta}(0)}{\sqrt{nh_{n} ^{2d}}}+o_{\mathbb{P}}(1), \] \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{LR-BC,}\ast} ]-\hat{\theta}_{n}^{\mathtt{LR-BC}})=\frac{2K(0)-K^{\Delta}(0)}{\sqrt {nh_{n}^{2d}}}+o_{\mathbb{P}}(1), \] \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{LR-GJ,}\ast} ]-\hat{\theta}_{n}^{\mathtt{LR-GJ}})=o_{\mathbb{P}}(1), \] and \[ \sqrt{n}(\mathbb{E}_{n}^{\ast}[\hat{\theta}_{n}^{\mathtt{LR-LO,}\ast} ]-\hat{\theta}_{n}^{\mathtt{LR-LO}})=\frac{2K(0)-K^{\Delta}(0)}{\sqrt {nh_{n}^{2d}}}+o_{\mathbb{P}}(1). \]
As a consequence, $\hat{\theta}_{n}^{\mathtt{LR,}\ast}$ and $\hat{\theta} _{n}^{\mathtt{LR-GJ,}\ast}$ satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) $ under Conditions D, K, and B, whereas Condition B must be strengthened to Condition B$^{+}$ for $\hat{\theta}_{n}^{\mathtt{LR-BC,}\ast}$ and $\hat{\theta}_{n} ^{\mathtt{LR-LO,}\ast}$ to satisfy $\left( \ref{Bootstrap consistency: Bias condition}\right) .$