# Efficient Semiparametric Estimation of Average Treatment Effects Under
Covariate Adaptive Randomization

Ahnaf Rafi 111The author would like to thank Ivan Canay, Joel Horowitz, Eric
Auerbach and Federico Bugni. Their advice, suggestions and comments have been
invaluable in the development and writing of the paper. Additional thanks go
to Yong Cai, Deborah Kim, Filip Obradovic, Amilcar Velez as well as the
participants of the Spring 2021 and Spring 2022 Econometrics Reading Groups at
Northwestern University for their suggestions and comments.
Department of Economics
Northwestern University
<EMAIL_ADDRESS>
###### Abstract

Experiments that use covariate adaptive randomization (CAR) are commonplace in
applied economics and other fields. In such experiments, the experimenter
first stratifies the sample according to observed baseline covariates and then
assigns treatment randomly within these strata so as to achieve balance
according to pre-specified stratum-specific target assignment proportions. In
this paper, we compute the semiparametric efficiency bound for estimating the
average treatment effect (ATE) in such experiments with binary treatments
allowing for the class of CAR procedures considered in [10, 11]. This is a
broad class of procedures and is motivated by those used in practice. The
stratum-specific target proportions play the role of the propensity score
conditional on all baseline covariates (and not just the strata) in these
experiments. Thus, the efficiency bound is a special case of the bound in
[26], but conditional on all baseline covariates. Additionally, this
efficiency bound is shown to be achievable under the same conditions as those
used to derive the bound by using a cross-fitted Nadaraya-Watson kernel
estimator to form nonparametric regression adjustments.

Keywords: Efficient semiparametric estimation, average treatment effect,
randomized experiments, covariate adaptive randomization

JEL Classification: C14, C90

##  1 Introduction

Experiments that use covariate adaptive randomization (CAR) are commonplace in
applied economics and other fields. In such experiments, the experimenter
first groups sample units according to observed baseline covariates
(stratification) and then assigns treatment randomly to achieve _balance_
within these groups (strata). The term balance here means that the
experimenter additionally specifies stratum-specific target treatment
proportions and assigns treatment so that the proportion of units assigned to
treatment reaches the corresponding target as the sample size grows across
strata. A textbook treatment of covariate adaptive and stratified
randomization in clinical trials can be found in [51]. For review articles on
their use in development economics, see [20], [9] and [3]. In experiments
comparing outcomes from binary treatments, the quantity of interest is often
the average treatment effect (ATE). In this paper, we are concerned with
efficient semiparametric estimation of the ATE while allowing for a broad
class of CAR procedures motivated by those used in practice. We have two main
questions. The first is: is there a well-defined semiparametric efficiency
bound (SPEB) for this class of procedures? The second is: does there exist a
semiparametric estimator that achieves this bound and under what conditions
will this happen? Our answers to both are affirmative. For the first, we show
that a version of the bound in [26] is valid. For the second, we show that
under the same (weak) conditions used for derivation of the bound, there is a
feasible estimator that achieves the bound asymptotically.

In randomized experiments, correctly implemented randomization ensures that in
expectation, confounding factors are equally distributed across treatment arms
so that differences in outcomes are solely due to the differences in
treatment. Stratified randomization additionally aims to ensure that along
observable dimesions, this also remains true in practice. In CAR experiments,
both discrete and continuously distributed covariates are combined to form
strata. As shown in [3, Section 7.2 ], the main statistical benefit of
stratified randomization for ATE estimation is improved precision. The
standard recommendation for ATE estimation in stratified experiments is to
regress the observed outcomes on indicators for strata and their interactions
with treatment status in a linear regression equation (a fully saturated
linear regression model). The coefficients on the interaction terms from this
regression are then combined with sample stratum proportions to construct the
ATE estimate. The resulting estimator of the ATE is analogous to the Horvitz-
Thompson estimator ([31]) and does not use information beyond the strata.
However, experimenters concerned with estimation precision may want to use
information in the baseline covariates not contained in the strata.

An additional complication in the CAR context comes from the choice of
treatment assignment mechanism by the experimenter. Many popular treatment
assignment mechanisms used in practice aim for faster targeting of the target
assignment proportions than simple independent and identically distributed
(i.i.d.) assignment and in doing so, induce dependence in the observed
outcomes through dependence in the treatment assignments. One such example is
stratified permuted block randomization (SPBR, þ2.2). [10] provide additional
examples. The dependence induced by CAR designs can affect the behavior of ATE
estimators in surprising ways. Analyzing the case where target assignment
proportions are constant across strata, [10] show that the standard difference
in treatment and control group means can have a limit variance that depends
explicitly on the choice of treatment assignment mechanism. For example, all
else held equal, the same estimator exhibits a larger limit variance under
i.i.d. treatment assignments than when treatments are assigned according to
SPBR, even when both mechanisms have the same target proportions. [10] also
show that the same phenomenon holds true of the “stratum fixed effects”
estimator which is recommended by [9]. [11] extends this work and consider
both multiple treatments and target assignment proportions that vary by
strata. They show that the estimator of the ATE from a fully saturated
regression has a limit variance that depends on the target proportions (among
other things), but not on the particular choice of assignment mechanism. They
also show that for the stratum fixed effects estimator however, the limit
variance is still affected explicitly by the choice of assignment mechanism.
[10, 11] do not consider efficiency questions.

There is a literature that considers efficiency gains in experiments from
using baseline covariates via linear regression adjustments. However, whether
there are gains at all depends on the linear regression specification. In the
case without stratification, [23] shows that linear regression adjustments
(without interactions between treatment status and baseline covariates) can
hurt asymptotic precision of the ATE estimates though the estimates remain
consistent. However, work by [66] and [37] show that appropriately formed
regression adjustments (i.e. with the correct interaction terms) cannot hurt
(and indeed can improve) asymptotic precision in ATE estimation. For CAR
experiments, [38] build on the results of [10, 11] and show that correctly
formed linear regression adjustments cannot hurt (and can improve) asymptotic
precision under CAR. These works do not treat semiparametrically efficient
estimation.

In this paper, we are concerned with the semiparametrically efficient
estimation of the ATE in the CAR framework under the minimal set of
assumptions laid out by [11]. There is a large statistics literature around
semiparametric efficiency, starting with the seminal work of [58]. Most of
these are developed assuming i.i.d. data. [7] offers a comprehensive textbook
treatment and [41] provides an approachable review. Notable examples with
non-i.i.d. data can be found in [8], [24] and [35] as well in references
therein. For treatment effects, [26] derives the SPEB for the ATE in
observational studies with i.i.d. data when treatment assignment is ignorable
conditional on observable covariates (ignorable as in [50, Section 1.3 ]). The
[26] bound does not apply immediately to our setting since the treatment
assignment rule is allowed to depend on the entire profile of sample strata.
For instance, it is not clear _a priori_ if the choice of assignment mechanism
will affect the efficiency bound since it can clearly influence limit behavior
of estimators as in [10, 11]. Furthermore, even when treatment assignments are
i.i.d. in a CAR context, the covariates being conditioned on during assignment
are the strata, so it is again not immediate from the bound in [26] what role
the additional baseline covariates can play in providing efficiency gains.
This paper shows that a version of the [26] bound accounting for both stratum-
specific target proportions and all baseline covariates is the SPEB across all
CAR experimental procedures. The target proportions play the role of the
propensity score conditional on baseline covariates. Additionally, the choice
of assignment mechanism does not affect this bound, only the target
proportions do. We derive this by using the partial sums arguments of [10, 11]
to show that the log likelihood ratios under $1/\sqrt{n}$ local alternatives
has the local asymptotic normality (LAN) property of [36].

In concurrent work, [1] proves the LAN property under a larger class of
experimental designs using martingale methods. The experiments considered
there include the CAR framework and additionally allows for arbitrary
dependence of the treatment rule on the covariates as well as observations of
past outcomes (to account for sequential assignment). [1]’s main goal is to
show that an optimized form of the [26] bound is a lower bound on limit
variance of the ATE across all experimental designs in that class. This is
motivated by recent papers on optimal design of experiments by using past
waves to either allocate treatment sequentially (e.g. [27]) or to form optimal
strata in a main experiment (e.g. [61], [4] and [18]). The optimized bound is
analogous to implementing a “conditional on covariates” Neyman allocation
([44]). We differ from [1] work on two counts. First, while their lower bound
holds across the designs the aforementioned class, they do not provide
efficiency bounds in specific instances within the class. We provide
efficiency bounds for a given fixed stratification scheme and we do not
consider the question of optimizing the bound. As a result our efficiency
bound as a variance lower bound is higher than the optimized one in [1] and
hence sharper for a given fixed stratification scheme. Second, [1] does not
consider the question of when the bounds are achievable. We do this for the
CAR framework explicitly as explained in the subsequent two paragraphs.

Once an efficiency bound is established, the question of its sharpness
remains, in the sense of whether or not it is achievable. A well known
phenomenon in the literature is that conditions under which a SPEB is
achievable are typically much stronger than those required to derive the
bound. [46] provide counterexamples where a finite and non-singular SPEB
exists but in certain non-trivial submodels, even consistent (let alone
efficient) estimation of the parameter of interest is impossible. One of their
examples is the partially linear model ([21], [49]) which is commonly used in
applied economics. A well known condition in the literature is that
$1/\sqrt{n}$-consistent and asymptotically normal ($1/\sqrt{n}$-CAN) two-step
semiparametric estimators require any first step infinite-dimensional
(henceforth nonparametric) nuisance parameters to be estimated at a rate
faster than $n^{-\frac{1}{4}}$ where $n$ is the sample size (see for instance
[42] and [12]). This is mainly because nonparametric estimators exhibit
considerable bias and this condition limits the effect of this bias on the
second estimation step. The $n^{-\frac{1}{4}}$ rate condition however is
especially restrictive when the covariates are of higher dimension, due to the
curse of dimensionality. Achieving this rate condition typically requires
imposing smoothness and/or Donsker conditions (often dimension dependent) on
unknown nuisance parameters and the estimator in question. For the ATE, [26]
shows that semiparametrically efficient estimation can be done under
regularity conditions through nonparametric regression adjustments or
imputation. In their first step, series estimators of conditional means and
the propensity score are used. Estimation by kernel methods can also be done,
see [33] for a review. In all of these, the use of smoothness restrictions on
nonparametric population unknowns is ubiquitous.

An additional contribution of this paper is to show that the SPEB derived for
the ATE is achievable under the same conditions used for its derivation. We do
this by leveraging the _efficient influence function_ to form an estimating
equation (or moment condition) for the ATE. The resulting estimator is the
familiar (and famed) augmented inverse probability weighted (AIPW) estimators
due to [47], [48], and [54]. We further note that since the propensity score
is known in these experiments, the estimating equation is linear in the
unknown nonparametric component and has the local robustness property of [15]
and [17]. This reduces the first order impact of bias in the first stage
nonparametric estimates on the resulting ATE estimator considerably and allows
for efficient estimation under much weaker conditions. The nonparametric
unknowns here are conditional means of the potential outcomes given baseline
covariates. We show first that for a generic estimator of conditional means, a
weak $L_{2}$ consistency requirement combined with cross-fitting as in [15] is
sufficient for efficient estimation of the ATE under CAR. The conditions under
which the particular choice of estimator achieves this $L_{2}$ consistency are
first left abstract \- different conditions apply to kernel estimators, random
forests, series estimators and neural networks for instance. For a given
choice of nonparametric estimator, smoothness conditions or functional form
restrictions may be unavoidable in achieving the $L_{2}$ consistency property.
Next, we establish that if the particular nonparametric estimator is a cross-
fitted Nadaraya-Watson kernel regression estimator, then efficient estimation
is possible under no additional restrictions on the semiparametric model. Our
motivation for using the cross-fitted Nadaraya-Watson estimator is two-fold.
First, the results of [19] and [57] show that the Nadaraya-Watson estimator is
universally $L_{2}$ consistent if the outcome in the regression has a finite
second moment. Second, the algebra of the cross-fitted Nadaraya-Watson
estimator is convenient in showing negligiblility of remainder terms.

We provide simulation evidence of the finite sample performance of the
feasible efficient estimator. Our simulations compare this estimator to an
infeasible efficient estimator as well as an estimator that uses information
only from the stratum labels but not the additional baseline covariates. A
comparison is also provided with an imputation estimator of [14] and [26].
Across all simulations designs, we find that using the feasible efficient
estimator produces an efficiency gain of 13% in terms of mean squared error
reduction in comparison to the estimator that discards information from
baseline covariates beyond strata. The maximal gain in this comparison over
all simulations is a 40% reduction in mean squared error. Compared to the
imputation estimator, our estimator exhibits considerably less bias across
most simulation designs.

The remainder of the paper is organized as follows. Section 2 sets up the
assumptions about the underlying population from which the sample is drawn as
well as the sampling process and the treatment assignment scheme. Section 3
presents the main result concerning the semiparametric efficiency bound (þ3.1)
after a discussion about parametric submodels in CAR experiment context.
Section 4 shows that the efficiency bound is achievable under the same
assumptions as used for its derivation via a two-step semiparametric estimator
using the efficient influence function. Section 5 provides Monte Carlo
evidence of the performance of the efficient estimator in finite samples.
Section 6 concludes the paper.

##  2 Preliminaries

This section describes assumptions on the populations of interest as well as
the sampling process which produces the observed data. This is done in two
subsections. Subsection 2.1 first describes the unobserved study population of
interest through a Neyman-Rubin causal model. Then, a target _observed
population_ is described for all CAR experiments considered in this paper.
This target observed population is useful for deriving the efficiency bound in
Section 3. Next, Subsection 2.2 describes sampling assumptions and provides
the two main examples of sampling schemes in this paper.

Throughout this paper, all random variables and vectors will be defined on a
sufficiently rich underlying probability space $(\Omega,\mathscr{F},\Pr)$.
Independence of random elements is denoted by  $\models$ and expectations
computed against the probability measure $\Pr$ are denoted by
$\mathbb{E}[\cdot]$. We will seldom refer to the underlying probability space
but define it nonetheless for clarity. Furthermore, absent any subscripts,
$\mathbb{E}[\cdot]$ will also denote expected values assuming the “true”
distribution of underlying data to be defined later on. The $d$-dimensional
multivariate normal distribution with mean vector
$\mathbf{m}\in\mathbb{R}^{d}$ and covariance matrix
$\Sigma\in\mathbb{R}^{d\times d}$ is denoted
$\mathcal{N}\left(\mathbf{m},\Sigma\right)$. We denote the matrix/vector
transpose operation by $(\cdot)^{\prime}$. The natural numbers are denoted by
$\mathbb{N}=\\{1,2,3,4,\dots\\}$ and for a given $\mathcal{S}\in\mathbb{N}$,
denote
$\mathbb{N}_{\mathcal{S}}=\\{1,\dots,\mathcal{S}\\}=\\{s\in\mathbb{N}:1\leq
s\leq\mathcal{S}\\}$.

###  2.1 A population framework for CAR experiments.

We employ a standard binary treatment potential outcomes framework and assume
that samples are drawn from an infinite super-population. The population of
interest is described by a random vector $W$ taking values in
$\mathbb{R}^{2+k}$ for some $k\in\mathbb{N}$, with
$W^{\prime}=\left(Y(0),Y(1),Z^{\prime}\right)$. For each $a\in\\{0,1\\}$,
$Y(a)$ is a scalar random variable representing the potential outcome from
receiving treatment $a$. The treatment $a=1$ can be interpreted as an
“innovation” whereas $a=0$ is a “status quo” or “control”. $Z$ is a random
$\mathbb{R}^{k}$-vector of baseline covariates. We denote the true
distribution of $W$ by $Q_{0}$. Prior to the assignment of treatment, only
baseline covariates are observable. Furthermore, after treatment has been
assigned, only the outcome associated with received treatment is observable so
that $(Y(0),Y(1))$ is not jointly observable. We maintain the following
assumptions about the population distribution $Q_{0}$.

######  Assumption 2.1.

þ Let $\mu_{Z}$ be a $\sigma$-finite measure on the Borel sets of
$\mathbb{R}^{k}$ and for $a\in\\{0,1\\}$, let $\mu_{a}$ be a $\sigma$-finite
measure on the Borel sets of $\mathbb{R}$. Furthermore, let $\mu$ denote the
product measure on the Borel sets of $\mathbb{R}^{2+k}$ constructed from
$\mu_{0}$, $\mu_{1}$ and $\mu_{Z}$. The true population distribution, $Q_{0}$,
belongs to a family $\mathbf{Q}$ of distributions on $\mathbb{R}^{2+k}$ such
that for each $Q\in\mathbf{Q}$,

  1. (a)

$Q$ is dominated by $\mu$ with Radon-Nikodym density
$q\left(\cdot;Q\right)=\mathrm{d}Q/\mathrm{d}\mu$.

  2. (b)

The potential outcomes have finite second moments under $Q$, i.e.
$\mathbb{E}_{Q}\left[Y{(a)}^{2}\right]<\infty$ for each $a\in\\{0,1\\}$, where
$\mathbb{E}_{Q}$ denotes the expected value assuming data are distributed
according to $Q$.

The dominance assumption is made for mathematical convenience and is standard
in the literature on semiparametric efficiency. Additionally, the choices of
dominating measures $\mu_{0},\mu_{1}$ and $\mu_{Z}$ are irrelevant and do not
affect the derivation of the efficiency bound. The finite second moment
condition is used for deriving Gaussian limiting distributions in later
sections. The family of distributions $\mathbf{Q}$ is a nonparametric family
since it is infinite dimensional. The parameter of interest is the average
treatment effect (ATE), $\beta_{\ast}:\mathbf{Q}\to\mathbb{R}$ defined by

$\beta_{\ast}(Q)=\mathbb{E}_{Q}[Y(1)-Y(0)]=\int\left(y_{1}-y_{0}\right)\;Q(\mathrm{d}y_{0},\mathrm{d}y_{1},\mathrm{d}z).$
(2.1)

The true value of the ATE will be denoted
$\beta_{0}=\beta_{\ast}\left(Q_{0}\right)$.

In both observational and experimental studies on the average effect of
treatment, observed outcomes result from the receipt of treatment. That is,
observed outcomes are given by $Y$ defined by

$Y=Y(1)\cdot A+Y(0)\cdot(1-A)$

where $A$ is a Bernoulli random variable denoting the treatment received. In
experimental settings, the conditional distribution of $A$ given the baseline
covariates $Z$ is assumed to be fully known to the experimenter. The target
observed population in the special case of a CAR experiment is the
distribution of the vector $X^{\prime}=\left(Y,A,Z^{\prime}\right)$ which
satisfies þ2.2 below.

######  Assumption 2.2 (Simple Stratified Randomization Assignment).

þ There exists a $\mathcal{S}\in\mathbb{N}$, a measurable function
$\mathbb{S}:\mathbb{R}^{k}\to\mathbb{N}_{\mathcal{S}}$ and a vector
$\pi=(\pi(1),\dots,\pi(\mathcal{S}))$ with $\pi(s)\in(0,1)$ for every
$s\in\mathbb{N}_{\mathcal{S}}$, which are all known to the experimenter. The
observable random vector $X^{\prime}=(Y,A,Z^{\prime})$ is constructed from $W$
and its components satisfy

$\displaystyle Y=Y(1)\cdot A+Y(0)\cdot(1-A),$ (2.2)
$\displaystyle[(Y(0),Y(1),Z)\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}A]|\mathbb{S}(Z),$
(2.3) $\displaystyle[A|\mathbb{S}(Z)=s]\sim\mathrm{Bernoulli}(\pi(s)).$ (2.4)

The function $\mathbb{S}$ is used to construct a (measurable) finite partition
of the support of the covariates $Z$ and thus, $\mathcal{S}$ is an upper bound
on the number of strata. The condition in (2.3) requires that assignment to
treatment be exogenous to both potential outcomes and the remaining variation
in covariates given stratum labels. The condition in (2.4) requires that
treatment assignment proportions correspond to the pre-specified target
probabilities in $\pi$. In finite samples, the covariate adaptive
randomization schemes that will be described in the next subsection all
provide different ways to target the experiment in þ2.2.

######  Remark 2.1.

þ Let $\mathbf{P}$ denote the family of distributions for the random vector
$X$ that is determined by þ2.1 and þ2.2. Let $P_{0}$ denote the distribution
in $\mathbf{P}$ that corresponds to the true population distribution
$Q_{0}\in\mathbf{Q}$. It is straightforward to show that the ATE in (2.1) is
nonparametrically identifiable (see [40, Definition 3.2 ]) via the map
$\beta:\mathbf{P}\to\mathbb{R}$

$\beta(P)=\mathbb{E}_{P}\left[\frac{Y\cdot
A}{\pi(\mathbb{S}(Z))}-\frac{Y\cdot(1-A)}{1-\pi(\mathbb{S}(Z))}\right].$ (2.5)

That is, for a given $Q\in\mathbf{Q}$, if $P(Q)$ denotes the distribution in
$\mathbf{P}$ formed from $Q$ and þ2.2, then $\beta_{\ast}(Q)=\beta(P(Q))$. In
particular, the true value of the average treatment effect can be written as

$\beta_{0}=\beta_{\ast}\left(Q_{0}\right)=\beta\left(P_{0}\right).$ (2.6)

###  2.2 Sampling framework for CAR experiments.

In this subsection, we describe the assumptions maintained for the sampling
process that produces observed data in a CAR experiment.

######  Assumption 2.3.

þ $\mathbf{W}=\left\\{W_{i}:i\in\mathbb{N}\right\\}$ is a sequence of
independent and identically distributed (i.i.d.) random
$\mathbb{R}^{2+k}$-vectors with $W_{i}\sim Q_{0}$ for every $i\in\mathbb{N}$.
The first $n\in\mathbb{N}$ elements of $\mathbf{W}$, denoted by
$\mathbf{W}_{n}^{\prime}=\left(W_{1},\dots,W_{n}\right)$, are the potential
outcome and covariate values for observations in the sample.

þ2.3 states that potential outcome and covariate values for observations in
the sample are drawn at random from the population distribution $Q_{0}$. That
is, if both potential outcomes and covariates were all observable, the
experimenter would have a representative sample from the underlying
population. This assumption is maintained in the recent literature on CAR
experiments as well as optimal experimental designs, for instance in [10],
[11], [61], [4] and [18]. Given the baseline covariates and corresponding
strata, the experimenter chooses a vector of treatment assignments
$\mathbf{A}_{n}^{\prime}=\left(A_{n1},\dots,A_{nn}\right)$ which is a random
vector supported in ${\\{0,1\\}}^{n}$. The experimenter has full control over
the distribution of treatment assignments and treatment assignments do not
have to be i.i.d. We introduce the following notation for convenience. For
each $s\in\mathbb{N}_{\mathcal{S}}$ and $a\in\\{0,1\\}$ the stratum size and
stratum treatment group size are respectively

$N_{n}(s)=\sum_{i=1}^{n}\mathbb{I}\left\\{\mathbb{S}\left(Z_{i}\right)=s\right\\}\quad\text{and}\quad
N_{n}(a,s)=\sum_{i=1}^{n}\mathbb{I}\left\\{A_{ni}=a,\mathbb{S}\left(Z_{i}\right)=s\right\\}.$
(2.7)

We maintain the following assumptions about the observed data in a covariate
adaptive randomized experiment.

######  Assumption 2.4.

þ Let $\mathcal{S}$, $\mathbb{S}$ and $\pi$ be as in þ2.2. For a sample of
size $n\in\mathbb{N}$, the observed data are
$\mathbf{X}_{n}^{\prime}=\left(X_{n1},\dots,X_{nn}\right)$ with individual
observations $X_{ni}=\left(Y_{ni},A_{ni},Z_{i}\right)$ for each
$i\in\mathbb{N}_{n}$. The observed outcomes and treatment assignment mechanism
satisfy the following.

  1. (a)

The observed outcomes are given by

$Y_{ni}=Y_{i}(1)A_{ni}+Y_{i}(0)\left(1-A_{ni}\right).$ (2.8)

  2. (b)

For every $n\in\mathbb{N}$,
$\left[\mathbf{W}_{n}\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{A}_{n}\right]|\mathbf{S}_{n}$
where $\mathbf{S}_{n}^{\prime}=\left(S_{1},\dots,S_{n}\right)$ and
$S_{i}=\mathbb{S}\left(Z_{i}\right)$ for each $i\in\mathbb{N}_{n}$.

  3. (c)

For every $n\in\mathbb{N}$, the conditional distribution of treatment
assignments given the profile of sample strata

$\alpha_{n}\left(\mathbf{a}_{n}\middle|\mathbf{s}_{n}\right):=\Pr\left(\mathbf{A}_{n}=\mathbf{a}_{n}\middle|\mathbf{S}_{n}=\mathbf{s}_{n}\right)\qquad\forall\mathbf{a}_{n}\in\\{0,1\\}^{n},\forall\mathbf{s}_{n}\in\mathbb{N}_{\mathcal{S}}^{n}$
(2.9)

is completely known to the experimenter and does not depend on the population
distribution $Q_{0}$.

  4. (d)

Under $Q_{0}$ defined in þ2.1 and $\alpha_{n}(\cdot|\cdot)$ chosen by the
experimenter above, for each $s\in\mathbb{N}_{\mathcal{S}}$,

$\frac{N_{n}(1,s)}{N_{n}(s)}\overset{\mathrm{p}}{\to}\pi(s).$ (2.10)

Denote the distribution of $\mathbf{X}_{n}$ by $P_{0,n}$. Note that $P_{0,n}$
is determined by the population distribution $Q_{0}$, the stratification
scheme $\mathbb{S}$, equation (2.8), and the choice of randomization scheme.
þ2.4 places the same set of restrictions on $P_{0,n}$ as in [11] on the
relationship between the randomization scheme, the underlying population and
the strata. þ2.4 (a) requires that the observed outcome for a given
observation is exactly the potential outcome associated with the assigned
treatment (as in equation (2.2) for the target experiment population
$\mathbf{P}$). þ2.4 (b) requires that the treatment assignment be ignorable
(or exogenous) given the strata. In particular, the assignment scheme can only
be a function of the stratum labels and a randomization device exogenous to
the information contained in the potential outcomes and the baseline
covariates beyond that afforded by the strata. This assumption is analogous to
(2.3) in þ2.2. Its main use is to guarantee identification of the ATE within
each stratum and it is additionally a sufficient condition to identify the
overall ATE. þ2.4 (c) requires that the randomization procedure be fully known
to the experimenter. This assumption plays a role in providing a simple
characterization of the joint distribution of the observed data and allows us
to avoid mathematical complications when talking about parametric sub-models
during the discussion of semiparametric efficiency. Finally, þ2.4 (d) requires
the randomization procedure to reach the target treatment proportion within a
given stratum at least asymptotically in the sense of convergence in
probability. This is analogous to (2.4) in þ2.2. There are a number of
examples of randomization schemes which will satisfy the requirements imposed
by þ2.4. We provide two examples of such randomization schemes.

######  Example 2.1 (Simple Stratified Random Assignment (SSRA)).

þ Let the assignments, $\left\\{A_{i}\right\\}_{i=1}^{n}$, be i.i.d. Bernoulli
random variables such that $\Pr\left(A_{i}=1|S_{i}=s\right)=\pi(s)$ and
$[(Y_{i}(0),Y_{i}(1),Z_{i})\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}A_{i}]|S_{i}$.
The observations $X_{i}^{\prime}=\left(Y_{i},A_{i},Z_{i}^{\prime}\right)$
generated by this process form an i.i.d. sample from $P_{0}$ defined in þ2.2.
The conditional mass function $\alpha_{n}(\cdot|\cdot)$ here is

$\alpha_{n}\left(\mathbf{a}_{n}\middle|\mathbf{s}_{n}\right)=\prod_{i=1}^{n}\pi\left(s_{i}\right)^{a_{i}}\left(1-\pi\left(s_{i}\right)\right)^{1-a_{i}}\qquad\forall\mathbf{a}_{n}\in\\{0,1\\}^{n},\forall\mathbf{s}_{n}\in\mathbb{N}_{\mathcal{S}}^{n},\forall
n\in\mathbb{N}.$ (2.11)

þ2.4 (d) can be verified by appealing to the Strong Law of Large Numbers.

######  Example 2.2 (Stratified Permuted Block Randomization (SPBR)).

þ Denote the integer floor function by $\lfloor\cdot\rfloor$. Within stratum
$s$, let

$N_{n}(1,s)=\left\lfloor\pi(s)N_{n}(s)\right\rfloor\quad\text{and}\quad\frac{1}{c_{s,n}}=\binom{N_{n}(s)}{N_{n}(1,s)}.$

There are $c_{s,n}^{-1}$ distinct subsets (or blocks) of size $N_{n}(1,s)$
from the overall stratum which has size $N_{n}(s)$. We can choose one of these
blocks uniformly at random, i.e. each distinct block meeting the size
requirements gets assigned a probability mass of $c_{ns}$. þ2.4 (b) can be
satisfied by ensuring the randomization device used to choose the treatment
block is independent to any outcome and covariate information within strata.
Since $|N_{n}(1,s)-\pi(s)N_{n}(s)|\leq 1$ almost surely for every
$s\in\mathbb{N}_{\mathcal{S}}$ and
$N_{n}(s)\overset{\mathrm{a.s.}}{\to}\infty$ if $Q(\mathbb{S}(Z)=s)>0$, þ2.4
(d) is also satisfied. Additionally, for each
$\mathbf{a}_{n}\in\\{0,1\\}^{n}$,
$\mathbf{s}_{n}\in\mathbb{N}_{\mathcal{S}}^{n}$ and $n\in\mathbb{N}$,

$\alpha_{n}\left(\mathbf{a}_{n}\middle|\mathbf{s}_{n}\right)=\prod_{s=1}^{\mathcal{S}}\binom{\sum_{i=1}^{n}\mathbb{I}\left\\{s_{i}=s\right\\}}{\sum_{i=1}^{n}a_{i}\mathbb{I}\left\\{s_{i}=s\right\\}}^{-1}\mathbb{I}\left\\{\sum_{i=1}^{n}a_{i}\mathbb{I}\left\\{s_{i}=s\right\\}=\left\lfloor\pi(s)\sum_{i=1}^{n}\mathbb{I}\left\\{s_{i}=s\right\\}\right\rfloor\right\\}.$
(2.12)

Both examples satisfy the requirement imposed by þ2.4 (d) for approaching the
stratum-specific target proportions, but they do so at different rates of
convergence. The SSRA method provides convergence to the stratum-specific
targets at a $n^{-\frac{1}{2}}$ rate due to the Lindeberg-Lévy Central Limit
Theorem. That is,

$\left(\frac{N_{n}(1,s)}{N_{n}(s)}-\pi(s):s=1,\dots,\mathcal{S}\right)=O_{\mathrm{p}}\left(\frac{1}{\sqrt{n}}\right)\text{
under SSRA.}$

It can be shown using the Strong Law of Large Numbers that SPBR achieves the
same targeting property at a faster $n^{-1}$ rate, so that

$\left(\frac{N_{n}(1,s)}{N_{n}(s)}-\pi(s):s=1,\dots,\mathcal{S}\right)=O_{\mathrm{p}}\left(\frac{1}{n}\right)=o_{\mathrm{p}}\left(\frac{1}{\sqrt{n}}\right)\text{
under SPBR.}$

When a randomization scheme achieves this faster convergence property we say
that it achieves “strong balance”. Numerous other kinds of stratified
randomization techniques have been developed and analyzed in the literature on
randomized control trials — for instance Efron’s biased coin design and Wei’s
urn — see [10] and references therein for detailed descriptions. Furthermore,
the treatment assignments are not required to be i.i.d. and hence, the
observed outcomes are all potentially non-i.i.d. For instance, with the SPBR
procedure illustrated in þ2.2, assignments are independent across strata, but
they exhibit dependence within strata. Indeed, most (if not all) assignment
schemes that achieve strong balance will result in treatment assignments and
observed outcomes that are non-i.i.d. However, we will show that the same
efficiency bound as that for the i.i.d. sampling scheme in þ2.1 holds for all
CAR schemes that satisfy þ2.4. Finally while the assumptions are satisfied by
a fairly broad class of randomization schemes, there are ones that violate
them that are used in practice, e.g. the minimization method of [45].
Developing efficiency theory for these methods may be of interest but they are
not covered in this work.

##  3 Semiparametric efficiency bound

In this section, we first state the semiparametric efficiency bound for the
ATE and discuss it briefly. We also discuss the derivation of the bound in two
subsections. As stated in the introduction, of the efficiency bounds
established in the literature, the closest one to our setting is that of [26].
This is derived under i.i.d. observational data under the assumption of
ignorable treatment assignment conditional on observable covariates ([50,
Section 1.3 ]). The [26] bound does not apply immediately to our setting since
the treatment assignment rule is allowed to depend on the entire profile of
sample strata. For instance, it is not clear _a priori_ if the choice of
assignment mechanism will affect the efficiency bound since it can clearly
influence limit behavior of estimators as in [10, 11]. Furthermore, even when
treatment assignments are i.i.d. in a CAR context, the covariates being
conditioned on during assignment are the strata, so it is again not immediate
from the bound in [26] what role the additional baseline covariates can play
in providing efficiency gains. The SPEB derived here provides an answer to
these questions. Before stating our bound, let
$m\left(a,z;Q\right)=\mathbb{E}_{Q}[Y(a)|Z=z]$ for each $a\in\\{0,1\\}$, and
for brevity, let $m_{\ast}(a,z)=m\left(a,z;Q_{0}\right)$ Furthermore, define

$\mathbb{V}_{\ast}=\mathbb{E}\left[\frac{\mathrm{Var}[Y(1)|Z]}{\pi(\mathbb{S}(Z))}+\frac{\mathrm{Var}[Y(0)|Z]}{1-\pi(\mathbb{S}(Z))}+\left\\{m_{\ast}(1,Z)-m_{\ast}(0,Z)-\beta_{0}\right\\}^{2}\right].$
(3.1)

The following theorem establishes that $\mathbb{V}_{\ast}$ above is the SPEB.

######  Theorem 3.1.

þ Under þLABEL:asm--Q,asm--iid-Q,asm--treat-strat, the semiparametric
efficiency bound for estimating the ATE from a CAR experiment is
$\mathbb{V}_{\ast}$ in (3.1).

The proof of þ3.1 provides formal justification for $\mathbb{V}_{\ast}$ as an
efficiency bound by appealing to extensions of the famed convolution ([28])
and local asymptotic minimax theorems ([29]) to semiparametric problems. The
efficiency bound in (3.1) is that of [26] for i.i.d. data, if treatment is
assigned conditional on baseline covariates according to the propensity score
$\pi(\mathbb{S}(\cdot))$. The intuition for this is that even though
randomization happens conditional on strata, the baseline covariates are
independent to treatment assignment after conditioning on strata (þ2.4 (b)).
Thus, the minimum possible limit variance in estimating the ATE can be
achieved by utilizing any additional information the baseline covariates may
have about potential outcomes. This can be better understood by considering
what happens if only information from the strata are used during estimation.
For instance, consider the fully saturated regression estimator of the ATE
from [11]. This estimator is constructed from regressing observed outcomes
$Y_{ni}$ on the stratum indicators and their interactions with treatment
status $A_{ni}$. The coefficients from this regression are combined to then
form the ATE estimate. The resulting estimator has a “weighted difference in
means” form:

$\widehat{\beta}_{n,\mathrm{SAT}}=\sum_{s=1}^{\mathcal{S}}\frac{N_{n}(s)}{n}\cdot\left[\frac{\sum_{i=1}^{n}Y_{ni}\cdot
A_{ni}\cdot\mathbb{I}\left(S_{i}=s\right)}{N_{n}(1,s)}-\frac{\sum_{i=1}^{n}Y_{ni}\cdot\left(1-A_{ni}\right)\cdot\mathbb{I}\left(S_{i}=s\right)}{N_{n}(0,s)}\right].$
(3.2)

[11] show that $\widehat{\beta}_{n,\mathrm{SAT}}$ is asymptotically normal
with mean-zero and limit variance given by

$\mathbb{V}_{\mathrm{SAT}}=\mathbb{E}\left[\frac{\mathrm{Var}[Y(1)|\mathbb{S}(Z)]}{\pi(\mathbb{S}(Z))}+\frac{\mathrm{Var}[Y(0)|\mathbb{S}(Z)]}{[1-\pi(\mathbb{S}(Z))]}+\left\\{\mathbb{E}[Y(1)|\mathbb{S}(Z)]-\mathbb{E}[Y(0)|\mathbb{S}(Z)]-\beta_{0}\right\\}^{2}\right].$
(3.3)

Our results thus establish that the fully saturated regression estimator of
the ATE achieves the SPEB among all estimators that only use information from
the stratum labels. This follows from comparing (3.1) and (3.3) and noting
that we have $\mathbb{V}_{\mathrm{SAT}}=\mathbb{V}_{\ast}$ if
$Z=\mathbb{S}(Z)$ (up to information preserving relabelling of strata). The
latter condition amounts to only using information given in stratum labels. An
additional implication is that when conditional means or conditional variances
of the potential outcomes given baseline covariates exhibit variation within
strata, $\mathbb{V}_{\ast}$ can be a strict improvement over
$\mathbb{V}_{\mathrm{SAT}}$ \- an observation also made by [26]. This is
confirmed by Monte Carlo simulation evidence in Section 5 in which
$\mathbb{V}_{\ast}$ can be approximately half of $\mathbb{V}_{\mathrm{SAT}}$
in some simulation designs (i.e. there is up to a 50% possible reduction in
asymptotic variance).

We provide a few additional comments on the SPEB in (3.1). Note that the
choice of assignment mechanism only affects the bound through the choice of
the target assignment proportions in $\pi(\cdot)$. All else held equal, the
SPBR assignment mechanism in þ2.2, or any other procedure satisfying þ2.4,
will produce the same SPEB as i.i.d. draws from $\mathbf{P}$ (þ2.1).
Furthermore, [26] notes that knowledge of the propensity score is ancillary to
the SPEB for the ATE. In the context of CAR, we will show that this knowledge
is useful for achieving the SPEB during estimation. An important intermediate
product of our derivation for this purpose is the _efficient influence
function_ , $\varphi_{0}$, which is defined by

$\varphi_{0}(y,a,z)=\frac{a}{\pi(\mathbb{S}(z))}\left[y-m_{\ast}(1,z)\right]-\frac{1-a}{1-\pi(\mathbb{S}(z))}\left[y-m_{\ast}(0,z)\right]+\left[m_{\ast}(1,z)-m_{\ast}(0,z)-\beta_{0}\right].$
(3.4)

$\varphi_{0}$ is called the efficient influence function because
$\mathbb{E}\left[\varphi_{0}(Y,A,Z)\right]=0$ and its variance is the SPEB in
(3.1), i.e. $\mathbb{E}\left[\varphi_{0}(Y,A,Z)^{2}\right]=\mathbb{V}_{\ast}$.
This function is a key ingredient to achieving the efficiency bound as
illustrated in Section 4. In the subsequent two subsections, we provide some
additional discussion of how (3.1) is derived, but mathematical details are
left to the appendix.

###  3.1 A product characterization of distribution of observed data

We establish here that the joint distribution of the observed data
$\mathbf{X}_{n}$, $P_{0,n}$, has a product structure even though outcomes and
treatment assignments are potentially non-i.i.d. This product structure will
then imply that $P_{0,n}$ has a density that also has a product form. To that
end, we first define a dominating measure for a single observation. For Borel
sets $\mathcal{Y}\subseteq\mathbb{R}$ and $\mathcal{Z}\subseteq\mathbb{R}^{k}$
and for $\mathcal{A}\subseteq\\{0,1\\}$, define the measures $\nu_{a}$ with
$a\in\\{0,1\\}$ and $\nu$ by

$\begin{split}\nu_{a}(\mathcal{Y}\times\mathcal{Z})=&\
\mu_{a}(\mathcal{Y})\cdot\mu_{Z}(\mathcal{Z}),\\\
\nu(\mathcal{Y}\times\mathcal{A}\times\mathcal{Z})=&\
\nu_{1}(\mathcal{Y}\times\mathcal{Z})\cdot\mathbb{I}(1\in\mathcal{A})+\nu_{0}(\mathcal{Y}\times\mathcal{Z})\cdot\mathbb{I}(0\in\mathcal{A}).\end{split}$
(3.5)

Let $q_{a}(y,z;Q)$ denote the marginal density of $(Y(a),Z)$ if the underlying
population has distribution $Q\in\mathbf{Q}$. Furthermore, let
$P_{n}(\cdot;Q)$ denote the distribution of the observed data $\mathbf{X}_{n}$
if $\mathbf{W}_{n}$ is an i.i.d. sample from $Q\in\mathbf{Q}$. Under this
notation, we have $P_{0,n}\equiv P_{n}\left(\cdot;Q_{0}\right)$.

######  Lemma 3.1.

þ Let $\mathbf{Y}_{n}^{\prime}=\left(Y_{n1},\dots,Y_{nn}\right)$ and
$\mathbf{Z}_{n}^{\prime}=\left(Z_{1},\dots,Z_{n}\right)$. For each
$\mathbf{y}_{n}\in\mathbb{R}^{n}$, $\mathbf{a}_{n}\in{\\{0,1\\}}^{n}$,
$\mathbf{s}_{n}\in{\mathbb{N}_{\mathcal{S}}}^{n}$ and
$\mathbf{z}_{n}^{\prime}=\left(z_{1},\dots,z_{n}\right)\in\mathbb{R}^{k\times
n}$, denote the event

$E_{n}\left(\mathbf{y}_{n},\mathbf{a}_{n},\mathbf{z}_{n},\mathbf{s}_{n}\right)=\left\\{\mathbf{Y}_{n}\leq\mathbf{y}_{n},\mathbf{A}_{n}=\mathbf{a}_{n},\mathbf{Z}_{n}\leq\mathbf{z}_{n},\mathbf{S}_{n}=\mathbf{s}_{n}\right\\}.$

where the inequalities are understood to be element-wise. Then under þ2.1,
þ2.2, þ2.4 and (2.8), if $\mathbf{W}_{n}$ is an i.i.d. sample from
$Q\in\mathbf{Q}$,

$P_{n}\left(E_{n}\left(\mathbf{y}_{n},\mathbf{a}_{n},\mathbf{z}_{n},\mathbf{s}_{n}\right);Q\right)=\
\alpha_{n}\left(\mathbf{a}_{n}\middle|\mathbf{s}_{n}\right)\times\prod_{i=1}^{n}\left\\{\begin{array}[]{l}Q\left(Y_{i}(1)\leq
y_{i},Z_{i}\leq z_{i},S_{i}=s_{i}\right)^{a_{i}}\\\ \times Q\left(Y_{i}(0)\leq
y_{i},Z_{i}\leq z_{i},S_{i}=s_{i}\right)^{1-a_{i}}\end{array}\right\\}.$ (3.6)

Thus, $P_{n}(\cdot;Q)$ is absolutely continuous against the $n$-fold product
measure formed from $\nu$ in (3.5) with density

$p_{n}\left(\mathbf{y}_{n},\mathbf{a}_{n},\mathbf{z}_{n};Q\right)=\alpha_{n}\left(\mathbf{a}_{n}\middle|\mathbf{s}_{n}\right)\prod_{i=1}^{n}q_{1}\left(y_{i},z_{i};Q\right)^{a_{i}}q_{0}\left(y_{i},z_{i};Q\right)^{1-a_{i}}.$
(3.7)

where in (3.7), we restrict
$\mathbf{s}_{n}^{\prime}=\left(\mathbb{S}\left(z_{1}\right),\dots,\mathbb{S}\left(z_{n}\right)\right)$.

The main consequence of þ3.1 is as follows. Let
$\mathcal{P}=\\{\mathbf{P}_{n}:n\in\mathbb{N}\\}$ be the sequence of families
of distributions for the observed data $\mathbf{X}_{n}$ such that for each
$n\in\mathbb{N}$, every member of $\mathbf{P}_{n}$ is determined by an i.i.d.
sample from some distribution in $\mathbf{Q}$ (as defined in þ2.1), the
observed outcome equation (2.8) and a treatment assignment mechanism that
satisfies þ2.4. Any member of $\mathbf{P}_{n}$ must then satisfy the product
structure in (3.6) and (3.7). Furthermore, different covariate adaptive
randomization schemes give rise to different sequences of families
$\mathcal{P}$ that only vary according to the choice of the sequence of
conditional treatment assignment distributions
$\left\\{\alpha_{n}(\cdot):n\in\mathbb{N}\right\\}$.

######  Remark 3.1 (Implications for Simple Stratified Random Assignment).

þ Consider the family $\mathbf{P}$ defined by þ2.2 in þ2.1. An immediate
additional consequence of þ3.1 is that any distribution $P\in\mathbf{P}$ has a
density against $\nu$ of the form

$p(y,a,z;P)=\left[q_{1}(y,z)\cdot\pi(\mathbb{S}(z))\right]^{a}\left[q_{0}(y,z)\cdot\pi(1-\mathbb{S}(z))\right]^{1-a}.$

Recall also that in þ2.1, we consider a treatment assignment rule that
essentially produces an i.i.d. sample of size $n$ from a distribution in
$\mathbf{P}$. In this case, we would of course have that
$\mathbf{P}_{n}=\mathbf{P}^{n}$ where the latter is the set of all $n$-fold
product measures formed from some measure in $\mathbf{P}$.

###  3.2 Deriving the semiparametric efficiency bound

In this subsection, we first introduce and discuss definitions of parametric
submodels and regular parametric submodels which are essential concepts to the
derivation of the semiparametric efficiency bound. It is then shown in þ3.2
that the log likelihood ratios in regular parametric submodels under
$1/\sqrt{n}$ local alternatives exhibit the local asymptotic normality (LAN)
property of [36]. An efficiency bound is established for parametric submodels
(þ3.3). Then, via a discussion on differentiability of the ATE in regular
parametric submodels we provide an intuitive description of how the parametric
efficiency bound is extended to the semiparametric efficiency bound presented
in þ3.1.

######  Definition 3.1.

þ A parametric submodel of $\mathcal{P}$ is a sequence of families of
distributions,
$\mathcal{P}^{0}=\left\\{\mathbf{P}_{n}^{0}:n\in\mathbb{N}\right\\}$ that
satisfies the following conditions.

  1. (a)

For every $n\in\mathbb{N}$, $\mathbf{P}^{0}_{n}\subseteq\mathbf{P}_{n}$. That
is, every member, $P_{n}\in\mathbf{P}^{0}_{n}$ is determined by a population
distribution from a subfamily $\mathbf{Q}^{0}\subseteq\mathbf{Q}$ and
satisfies þ2.4.

  2. (b)

There is a $\Theta\subseteq\mathbb{R}^{d}$ (where $d\in\mathbb{N}$), and a map
$\theta\mapsto\left(q_{0}(\cdot;\theta),q_{1}(\cdot;\theta)\right)$ from
$\Theta$ to $(Y(a),Z)$-marginal densities of $\mathbf{Q}^{0}$ such that every
member of $\mathbf{P}^{0}_{n}$ has a density (as in þ3.1) of the form

$\displaystyle
p_{n}\left(\mathbf{y}_{n},\mathbf{a}_{n},\mathbf{z}_{n};\theta\right)=$
$\displaystyle\
\alpha_{n}\left(\mathbf{a}_{n}|\mathbf{s}_{n}\right)\cdot\prod_{i=1}^{n}q_{1}\left(y_{i},z_{i};\theta\right)^{a_{i}}q_{0}\left(y_{i},z_{i};\theta\right)^{1-a_{i}},$
$\displaystyle\text{where }\mathbf{s}_{n}^{\prime}=$ $\displaystyle\
\left(\mathbb{S}\left(z_{1}\right),\dots,\mathbb{S}\left(z_{n}\right)\right)$

Additionally, the map
$\theta\mapsto\left(q_{0}(\cdot;\theta),q_{1}(\cdot;\theta)\right)$ satisfies
the $Z$-marginal restriction: there exists a $\mu_{Z}$-density function
$g(\cdot;\theta)$ (i.e. $g(z;\theta)\geq 0$ for all $z\in\mathbb{R}^{k}$ and
$\int g(z;\theta)\mu_{Z}(\mathrm{d}z)=1$) such that

$\int_{\mathbb{R}}q_{0}(y,z;\theta)\;\mu_{0}\left(\mathrm{d}y\right)=\int_{\mathbb{R}}q_{1}(y,z;\theta)\;\mu_{1}\left(\mathrm{d}y\right)=g(z;\theta)\quad\forall\theta\in\Theta\text{
for }z\in\mathbb{R}^{k}\ \mu_{Z}\text{-a.e.}$

We write $P_{\theta,n}$ for the element of $\mathbf{P}_{n}^{0}$ whose density
is $p_{n}(\cdot;\theta)$.

  3. (c)

The population subfamily $\mathbf{Q}^{0}$ contains the true distribution
$Q_{0}$ so that $P_{0,n}\in\mathbf{P}_{n}^{0}$. The parametrization
$\theta\mapsto p_{n}(\cdot;\theta)$ identifies the true distribution so that
there is a unique $\theta_{0}\in\Theta$ such that $P_{0,n}=P_{\theta_{0},n}$.

In þ3.1, condition (a) requires that a parametric submodel be a subset of the
overall semiparametric model. Condition (b) requires that the parametrization
$\theta\mapsto p_{n}(\cdot;\theta)$ produces densities of the same form as
those in þ3.1. The additional $Z$-marginal restriction ensures that both
densities $q_{0}(\cdot;\theta)$ and $q_{1}(\cdot;\theta)$ can be derived from
a common distribution $Q_{\theta}\in\mathbf{Q}^{0}$. We do not require the
submodel to specify the joint density of $(Y(0),Y(1),Z)$ since only the
marginals with respect to $(Y(0),Z)$ and $(Y(1),Z)$ are identified by the
randomized experiment (essentially due to (3.6) in þ3.1). Finally, (c)
requires the parametric submodel to contain the true distribution $P_{0,n}$.
Note that a parametric submodel does not parameterize the conditional mass
function of the treatment assignments, $\alpha_{n}$, since it is known and
does not depend on any population unknowns. This is different to the case of
observational data considered in [26] where the propensity score is unknown
and has to be parameterized.

The idea behind the use of parametric submodels is as follows. Since $\Theta$
(and therefore $\mathbf{P}_{n}$) in þ3.1 is finite dimensional, estimation of
the ATE restricted to $\mathcal{P}^{0}$ cannot be more difficult than
estimation within $\mathcal{P}$. One can imagine estimation of the ATE by
first estimating the nuisance parameter $\theta_{0}$ by maximum likelihood and
then forming a plug-in estimate of the ATE. If $\mathcal{P}^{0}$ is a
parametric submodel with a well-defined efficiency bound, any semiparametric
estimator for the ATE that is consistent and asymptotically normal under the
distributions in $\mathcal{P}$ cannot have asymptotic variance lower than the
efficiency bound under $\mathcal{P}^{0}$. Taking the supremum over all
parametric submodels, we get a variance lower bound for the semiparametric
model. The statement about parametric submodels having well-defined efficiency
bounds requires some qualification. Establishing efficiency requires
restricting our analysis to _regular_ parametric submodels (in which
efficiency bounds are well-defined) and to _regular_ estimators (to rule out
the phenomenon of super-efficiency). We provide the definition of a regular
parametric submodel in our case below. The definition of a regular estimator
can be found in [64, p. 115 and p. 365 ] or [63, p. 413 ].

######  Definition 3.2.

þ Let $\mathcal{P}^{0}$ be a parametric subfamily of $\mathcal{P}$ as in þ3.1.
$\mathcal{P}^{0}$ is called _regular_ at $\theta_{0}$ if the parametrization
$\theta\mapsto\left(q_{0}(\cdot;\theta),q_{1}(\cdot;\theta)\right)$ satisfies
the following.

  1. (a)

$\Theta$ is an bounded and open subset of $\mathbb{R}^{d}$.

  2. (b)

The maps $\theta\to\sqrt{q_{a}(\cdot;\theta)}$ are differentiable in the
quadratic mean at $\theta_{0}$. That is, for each $a\in\\{0,1\\}$, there is a
measurable function $D_{a}:\mathbb{R}^{1+k}\to\mathbb{R}^{d}$ such that $\int
D_{a}^{\prime}D_{a}\mathrm{d}\nu_{a}<\infty$ and

$\lim_{\|t\|\to
0}\|t\|^{-2}\int\left(\sqrt{q_{a}(y,z;\theta_{0}+t)}-\sqrt{q_{a}\left(y,z;\theta_{0}\right)}-D_{a}(y,z)^{\prime}\left(\theta-\theta_{0}\right)\right)^{2}\nu_{a}(\mathrm{d}y,\mathrm{d}z)=0.$

  3. (c)

The information matrix at $\theta_{0}$, $\mathcal{I}_{a}$, is non-singular,
where

$\mathcal{I}_{a}=4\int_{\mathbb{R}^{1+k}}D_{a}(y,z)D_{a}(y,z)^{\prime}\;\nu_{a}(\mathrm{d}y,\mathrm{d}z).$
(3.8)

The log-likelihood in any parametric submodel is

$\begin{split}\ell_{n}(\theta)=&\ \log
p_{n}\left(\mathbf{X}_{n};\theta\right)=\log\alpha_{n}\left(\mathbf{A}_{n}\middle|\mathbf{S}_{n}\right)+\sum_{i=1}^{n}\ell\left(Y_{ni},A_{ni},Z_{i};\theta\right),\\\
\text{where }\ell\left(Y_{ni},A_{ni},Z_{i};\theta\right)=&\ A_{ni}\log
q_{1}\left(Y_{ni},Z_{i};\theta\right)+\left(1-A_{ni}\right)\log
q_{0}\left(Y_{ni},Z_{i};\theta\right).\end{split}$ (3.9)

When the parametric submodel is regular, the associated score function at the
truth, $\theta_{0}$, is

$\begin{split}\dot{\ell}_{n}=&\
\sum_{i=1}^{n}\dot{\ell}\left(Y_{ni},A_{ni},Z_{i}\right),\\\ \text{where
}\dot{\ell}\left(Y_{ni},A_{ni},Z_{i}\right)=&\
A_{ni}\dot{\ell}_{1}\left(Y_{ni},Z_{i}\right)+\left(1-A_{ni}\right)\dot{\ell}_{0}\left(Y_{ni},Z_{i}\right),\\\
\dot{\ell}_{a}(y,z)=&\
2\frac{D_{a}(y,z)}{\sqrt{q_{a}\left(y,z;Q_{0}\right)}}\cdot\mathbb{I}\left\\{q_{a}\left(y,z;Q_{0}\right)>0\right\\}.\end{split}$
(3.10)

Note that the score in (3.10) is expressed in “root density” terms since it is
technically defined by the quadratic mean differentiability assumption in
þ3.2.222When the parametric submodels are such that the densities are
continuously differentiable with respect to $\theta$, then the usual log-
derivative form and (3.10) coincide. That is, we can write
$\dot{\ell}_{a}(\cdot)=\nabla_{\theta}\left[\log
q_{a}\right]\left(\cdot;\theta_{0}\right)$. Let $X^{\prime}=(Y,A,Z)$ be as in
the CAR target experiment defined by þ2.2. Under CAR, the associated Fisher
information matrix will be shown to be

$\begin{split}\mathcal{I}=&\
\mathbb{E}\left[\dot{\ell}(Y,A,Z)\dot{\ell}(Y,A,Z)^{\prime}\right]\\\ =&\
\sum_{s=1}^{\mathcal{S}}Q_{0}(\mathbb{S}(Z)=s)\left\\{\begin{array}[]{l}\pi(s)\mathbb{E}\left[\dot{\ell}_{1}(Y(1),Z)\dot{\ell}_{1}(Y(1),Z)^{\prime}\middle|\mathbb{S}(Z)=s\right]\\\
+(1-\pi(s))\mathbb{E}\left[\dot{\ell}_{0}(Y(0),Z)\dot{\ell}_{0}(Y(0),Z)^{\prime}\middle|\mathbb{S}(Z)=s\right]\end{array}\right\\}.\end{split}$
(3.11)

Note that the information matrix, $\mathcal{I}$ in (3.11) is related to the
marginal information matrices $\mathcal{I}_{a}$ for $a\in\\{0,1\\}$ in (3.8)
but adjusts these for stratification. Furthermore the score $\dot{\ell}$ and
information matrix $\mathcal{I}$ both depend on the choice of parametric
submodel since the quadratic mean derivatives $D_{a}$ depend on the choice of
parametric submodel. þ3.2 below shows that the main consequence of þ3.2 is
that the likelihood ratio in any regular parametric submodel has the usual
quadratic expansion and exhibits the local asymptotic normality (LAN) property
of [36] under general CAR procedures.

######  Lemma 3.2.

þ Suppose the parametric submodel $\mathcal{P}^{0}$ of $\mathcal{P}$ is
regular at the true value $\theta_{0}$. Then, $\mathcal{P}^{0}$ has the local
asymptotic normality property of [36] at $\theta_{0}\in\Theta$. In particular,
for $t\in\mathbb{R}^{d}$ such that $\theta_{0}+n^{-\frac{1}{2}}t\in\Theta$ for
all $n\in\mathbb{N}$, let the quadratic remainder
$R_{n}\left(\theta_{0},t\right)$ be defined by

$\ell_{n}\left(\theta_{0}+n^{-\frac{1}{2}}t\right)-\ell_{n}\left(\theta_{0}\right)=t^{\prime}\frac{1}{\sqrt{n}}\dot{\ell}_{n}-\frac{1}{2}t^{\prime}\mathcal{I}t+R_{n}\left(\theta_{0},t\right)$
(3.12)

where $\dot{\ell}_{n}$ is as in (3.10) and $\mathcal{I}$ is defined as in
(3.11). Let $G\sim\mathcal{N}\left(\mathbf{0}_{d},\mathcal{I}\right)$. Then
the following hold.

  1. (a)

Under $P_{0,n}$, $n^{-\frac{1}{2}}\dot{\ell}_{n}\overset{\mathrm{d}}{\to}G$.

  2. (b)

For any given $M>0$,
$R_{n}\left(\theta_{0},t\right)\overset{\mathrm{p}}{\to}0$ uniformly over
$\|t\|\leq M$ under $P_{0,n}$. That is, for any $\varepsilon>0$,

$\lim_{n\to\infty}\sup_{\|t\|\leq
M}P_{0,n}\left(\left|R_{n}\left(\theta_{0},t\right)\right|>\varepsilon\right)=0.$
(3.13)

The proof of þ3.2 in the appendix of this paper uses the tools developed [10]
and [11] to adapt the proof of Proposition 2.1.2 in the appendix of [7] to the
context of CAR. In particular, a partial-sums empirical process argument is
used to establish conclusion (a) in þ3.2. Using similar arguments, the sample
information matrix is shown to converge in probability to $\mathcal{I}$. The
remainder $R_{n}\left(\theta_{0},t\right)$ summarizes both the error replacing
the sample information matrix with its population (or limiting) analogue as
well as the error inherent in a second-order Taylor expansion. The latter is
shown to be negligible as well. Under the LAN phenomenon, þ3.3 below
establishes an efficiency result for estimation of the nuisance parameter
$\theta_{0}$.

######  Lemma 3.3.

þ Suppose $\mathcal{P}^{0}$ is a parametric submodel of $\mathcal{P}$ that is
regular at $\theta_{0}$. Let $\mathcal{I}$ be the Fisher information matrix
defined in (3.11). Then $\mathcal{I}$ is non-singular and $\mathcal{I}^{-1}$
is the efficiency bound for estimation of $\theta_{0}$. In particular, let
$G\sim\mathcal{N}\left(\mathbf{0}_{d},\mathcal{I}^{-1}\right)$. Then the
following hold.

  1. (a)

If $\widehat{\theta}_{n}$ is a sequence of regular estimators of $\theta_{0}$,
then

$\sqrt{n}\left(\widehat{\theta}_{n}-\theta_{0}-\frac{t}{\sqrt{n}}\right)\overset{\mathrm{d}}{\to}G+U.$
(3.14)

where $U$ is a random $\mathbb{R}^{d}$-vector independent to $G$ that is
specific to the estimator sequence $\widehat{\theta}_{n}$.

  2. (b)

Let $\mathcal{L}:\mathbb{R}^{d}\to\mathbb{R}$ be a bowl-shaped loss function,
i.e. $\mathcal{L}$ is non-negative, satisfies $\mathcal{L}(y)=\mathcal{L}(-y)$
and has convex sublevel sets. If $\widehat{\theta}_{n}$ is any sequence of
estimators of $\theta_{0}$, then

$\lim_{M\to\infty}\liminf_{n\to\infty}\
\sup_{\theta_{n}\in\Theta,\sqrt{n}\left\|\theta_{n}-\theta_{0}\right\|\leq
M}\mathbb{E}_{P_{\theta_{n},n}}\left[\mathcal{L}\left(\sqrt{n}\left(\widehat{\theta}_{n}-\theta_{n}\right)\right)\right]\geq\mathbb{E}\left[\mathcal{L}\left(G\right)\right].$
(3.15)

þ3.3 establishes that the usual Cramér-Rao parametric information bound holds
for estimation of $\theta_{0}$ in regular parametric submodels. This is true
even under general CAR procedures that may result in dependence within
observed data. þ3.3 is a direct consequence of the LAN property established in
þ3.2. In þ3.3, conclusion (a) follows from the celebrated convolution theorem
of [28] and conclusion (b) follows from the local asymptotic minimax theorem
of [29]. These are standard tools used to justify the parametric Cramér-Rao
bound. Note that in þ3.3 the random vectors $G$ and $U$ are again specific to
the parametric submodel chosen and in the case of $U$, also specific to the
choice of estimator sequence.

þ3.3 establishes an efficiency bound for estimation of the nuisance parameter
$\theta$ whereas the content of þ3.1 is about estimation of the ATE, $\beta$
in (2.5). To relate þ3.3 to a version of þ3.1, we have to first show that
$\beta$ is a pathwise differentiable parameter (see [7, Definition 3.3.1, p.
57 ] or [41, Section 3 ]). We do this by considering regular parametric
submodels of the target CAR experiment, $\mathbf{P}$, as defined by þ2.2 and
þ2.1. For a regular parametric submodel
$\mathbf{P}_{0}=\left\\{P_{\theta}:\theta\in\Theta\right\\}$ of $\mathbf{P}$,
the average treatment effect parameter is

$\gamma(\theta):=\beta\left(P_{\theta}\right)=\int_{\mathbb{R}^{1+k}}y\cdot
q_{1}(y,z;\theta)\;\nu_{1}(\mathrm{d}y,\mathrm{d}z)-\int_{\mathbb{R}^{1+k}}y\cdot
q_{0}(y,z;\theta)\;\nu_{0}(\mathrm{d}y,\mathrm{d}z).$

Adapting the pathwise differentiability arguments in [26] to the context of a
CAR experiment, the derivative (gradient) of $\gamma(\theta)$ at $\theta_{0}$
in any given regular parametric submodel can be written as

$\nabla\gamma\left(\theta_{0}\right)=\mathbb{E}\left[\varphi_{0}(Y,A,Z)\cdot\dot{\ell}(Y,A,Z)\right],$
(3.16)

where $\varphi_{0}$ is the efficient influence function from (3.4). The full
derivation of this adaptation is included the appendix for completeness
(þLABEL:lem--tangent-space-P,lem--pathwise-differentiability-ate). Combining
þ3.3 and (3.16), it follows via the usual “delta method style” argument that
the information bound for estimating the ATE in the regular parametric
submodel $\mathcal{P}^{0}$ is

$\mathbb{V}\left(\mathcal{P}^{0}\right)=\mathbb{E}\left[\varphi_{0}(Y,A,Z)\dot{\ell}\left(Y,A,Z\right)^{\prime}\right]\mathbb{E}\left[\dot{\ell}\left(Y,A,Z\right)\dot{\ell}\left(Y,A,Z\right)^{\prime}\right]^{-1}\mathbb{E}\left[\dot{\ell}\left(Y,A,Z\right)\varphi_{0}(Y,A,Z)\right].$

The variant of Cauchy-Schwarz inequality for random vectors established in
[62] can be used to conclude that
$\mathbb{V}\left(\mathcal{P}^{0}\right)\leq\mathbb{E}\left[\varphi_{0}(Y,A,Z)^{2}\right]=\mathbb{V}_{\ast}$.
This establishes that $\mathbb{V}_{\ast}$ in (3.1) is an upper bound over all
parametric information bounds for estimation of the ATE. However, one still
needs to show that $\mathbb{V}_{\ast}$ is in fact a supremum. This final step
is done by establishing that $\varphi_{0}$ can be approximated by the score
functions of regular parametric submodels.

##  4 Semiparametrically efficient estimation of the ATE under CAR

The efficiency bound derived in þ3.1 is a lower bound on asymptotic variances
of regular semiparametric estimators for the ATE. However, the question
remains as to whether it can be achieved. Typically, the conditions under
which estimators can achieve semiparametric efficiency bounds are stronger
than those required to derive the bounds. [46] provide examples for cases when
an efficiency bound can be established and shown to be finite as well as non-
singular, but even consistent estimation (let alone efficient) is impossible
without restricting the semiparametric model. One of their examples is the
partially linear model ([21], [49]). The restrictions they require are in the
form of additional smoothness conditions for the nonparametric nuisance
parameters that appear in the efficient influence function. These smoothness
conditions are to ensure that the nonparametric nuisance parameters can be
estimated consistently with a rate of convergence of at least $n^{-1/4}$ (see
also [42]). Furthermore, achieving $n^{-1/4}$-consistency for a non-parametric
estimator becomes more difficult when there are many covariates due to the
curse of dimensionality, and thus higher order (i.e. stronger) smoothness
conditions are required.

In the context of estimating average treatment effects with the assumption of
selection on observables (or ignorability conditional on covariates) and
i.i.d. data, [26] proposes nonparametric imputation estimators. These
estimators impute the potential outcome $Y_{i}(a)$ whenever it is unobserved
by using an estimate of the predicted value $m\left(a,Z_{i}\right)$, say
$\widehat{m}_{n}\left(a,Z_{i}\right)$. The difference of the imputed potential
outcomes is then averaged to estimate the ATE. [26] suggests the use of series
estimators for the propensity score $\Pr(A=1|Z=z)$ and the conditional
expectations $\mathbb{E}[Y\mathbb{I}\\{A=a\\}|Z=z]$ which then get combined to
construct estimators for $m_{\ast}(a,\cdot)$. Similar estimators appear in
various parts of the literature on semiparametric estimation of treatment
effects under treatment ignorability - see [33] for a review and examples with
kernel estimators for the aforementioned propensity score and conditional
expectations. The requirement of higher order smoothness conditions on the
propensity score and the conditional expectations $m_{\ast}(a,\cdot)$ are
ubiquitous in these examples.

In our case, the propensity score does not need to be estimated since the
target proportions by strata, $(\pi(1),\dots,\pi(\mathcal{S}))$, are chosen by
the experimenter. Furthermore, we should be able to use sample treatment
proportions by strata in place of true target proportions without any issue
since these form a finite-dimensional additional nuisance parameter. This
feature makes the efficient influence function linear in the nonparametric
nuisance parameters $m_{\ast}(a,\cdot)$. It is also straightforward to verify
that $\varphi$ has the Neyman orthogonality or local robustness property of
[15] and [16] with respect to estimation of $m_{\ast}(a,\cdot)$. To see this,
denote

$\begin{split}\varphi\left(y,a,z;\widetilde{m}(0,\cdot),\widetilde{m}(1,\cdot),b\right)=&\
\frac{a}{\pi(\mathbb{S}(z))}\left[y-\widetilde{m}(1,z)\right]-\frac{1-a}{1-\pi(\mathbb{S}(z))}\left[y-\widetilde{m}(0,z)\right]\\\
&+\left[\widetilde{m}(1,z)-\widetilde{m}(0,z)-b\right]\end{split}$ (4.1)

so that the efficient influence function satisfies
$\varphi_{0}(\cdot)=\varphi\left(\cdot;m_{\ast}(0,\cdot),m_{\ast}(1,\cdot),\beta_{0}\right)$.
If $m_{\ast}(0,\cdot),m_{\ast}(1,\cdot)$ were known, then we could estimate
$\beta$ by solving the sample equivalent of the moment condition

$\mathbb{E}\left[\varphi\left(Y,A,Z;m_{\ast}(0,\cdot),m_{\ast}(1,\cdot),\beta_{0}\right)\right]=0.$

The local robustness property stems from noting that at $\beta_{0}$,

$\frac{d}{d\tau}\mathbb{E}\left[\varphi\left(Y,A,Z;(1-\tau)m_{\ast}(0,\cdot)+\tau\widetilde{m}(0,\cdot),(1-\tau)m_{\ast}(1,\cdot)+\tau\widetilde{m}(1,\cdot),\beta_{0}\right)\middle]\right|_{\tau=0}=0.$

This local robustness property allows for $n^{-\frac{1}{2}}$ consistent and
efficient estimation of the ATE by using plug-in estimates of
$m_{\ast}(a,\cdot)$ under much weaker conditions. In particular, we do not
require smoothness conditions for $m_{\ast}(a,\cdot)$ to achieve the
efficiency bound in (3.1).

In the i.i.d. case (þ2.1), an “ideal” efficient estimator of the ATE would be
$\widetilde{\beta}^{\ast}_{n}=\beta_{0}+n^{-1}\sum_{i=1}^{n}\varphi_{0}\left(Y_{ni},A_{ni},Z_{i}\right)$.
Since adding $b$ to $\varphi$ in (4.1) removes it as an argument, this
estimator is

$\widetilde{\beta}^{\ast}_{n}=\frac{1}{n}\sum_{i=1}^{n}\left\\{\frac{A_{ni}\left[Y_{ni}-m_{\ast}\left(1,Z_{i}\right)\right]}{\pi\left(\mathbb{S}\left(Z_{i}\right)\right)}-\frac{\left(1-A_{ni}\right)\left[Y_{ni}-m_{\ast}\left(0,Z_{i}\right)\right]}{1-\pi\left(\mathbb{S}\left(Z_{i}\right)\right)}+m_{\ast}\left(1,Z_{i}\right)-m_{\ast}\left(0,Z_{i}\right)\right\\}.$
(4.2)

$\widetilde{\beta}_{n}^{\ast}$ is the familiar AIPW estimator of [47], [48],
and [54] if $m_{\ast}(a,\cdot)$ were known. A feasible estimator must replace
these with first stage estimates that can be computed from the sample. Since
it will be convenient later on, we allow for the estimates of
$m_{\ast}(a,\cdot)$ to vary across observations. We will also allow for the
experimenter to use estimated treatment proportions rather than the known true
ones. That is, let $\widehat{\pi}_{n}(s)$ be a consistent estimator for
$\pi(s)$ for each $s\in\mathbb{N}_{\mathcal{S}}$. A feasible estimator for
$\beta_{0}$ would be

$\begin{split}\widehat{\beta}^{\ast}_{n}=&\
\frac{1}{n}\sum_{i=1}^{n}\frac{A_{ni}\left[Y_{ni}-\widehat{m}_{ni}\left(1,Z_{i}\right)\right]}{\widehat{\pi}_{n}\left(\mathbb{S}\left(Z_{i}\right)\right)}-\frac{1}{n}\sum_{i=1}^{n}\frac{\left(1-A_{ni}\right)\left[Y_{ni}-\widehat{m}_{ni}\left(0,Z_{i}\right)\right]}{1-\widehat{\pi}_{n}\left(\mathbb{S}\left(Z_{i}\right)\right)}\\\
&+\frac{1}{n}\sum_{i=1}^{n}\left[\widehat{m}_{ni}\left(1,Z_{i}\right)-\widehat{m}_{ni}\left(0,Z_{i}\right)\right].\end{split}$
(4.3)

It is straightforward to show that $\widetilde{\beta}^{\ast}_{n}$ achieves the
efficiency bound $\mathbb{V}_{\ast}$ in (3.1) under the CAR procedures
satisfying our assumptions. Our plug-in estimators $\widehat{m}_{ni}$ will be
cross-fitted estimators as described in [15] (see their DML2 estimators). We
now provide conditions on the estimators $\widehat{m}_{ni}$, and
$\widehat{\pi}_{n,a}(s)$ as well as the overall model under which the
difference between $\widehat{\beta}^{\ast}_{n}$ to
$\widetilde{\beta}^{\ast}_{n}$ in (4.13) are asymptotically negligible after
scaling by $\sqrt{n}$. We start with assumptions on a sequence of
nonparametric estimators $\widehat{m}(n,a,\cdot;\cdot)$ from which
$\widehat{m}_{ni}(a,\cdot)$ will be constructed.

######  Assumption 4.1.

þ For $a\in\\{0,1\\}$,
$\widehat{m}\left(n,a,z;y_{1}(a),z_{1},\dots,y_{n}(a),z_{n}\right)$ is an
estimator sequence satisfying

$\lim_{n\to\infty}\mathbb{E}_{Q^{n+1}}\left[\left(\widehat{m}\left(n,a,Z;Y_{1}(a),Z_{1},\dots,Y_{n}(a),Z_{n}\right)-m(a,Z;Q)\right)^{2}\right]=0,$
(4.4)

when $Z$, $\left\\{Y_{i}(0),Y_{i}(1),Z_{i}\right\\}_{i=1}^{n}$ are i.i.d. with
distribution $Q\in\mathbf{Q}\left(\widehat{m}\right)$ with
$\mathbf{Q}\left(\widehat{m}\right)\subseteq\mathbf{Q}$. In the above,
$m(a,z;Q)=\mathbb{E}_{Q}\left[Y(a)|Z=z\right]$ and $\mathbb{E}_{Q^{n+1}}$ is
expectation under the product measure $Q^{n+1}$. Furthermore,
$Q_{0}\in\mathbf{Q}\left(\widehat{m}\right)$.

The estimator $\widehat{m}(n,a\cdot;\cdot)$ is allowed to use the entire
sample of potential outcomes for $a$ as well as the covariates. Of course, all
of these are not available - $\widehat{m}(n,a\cdot;\cdot)$ can only be
computed from the treatment group for treatment $a$. This will be accounted
for later on when we describe the cross-fitting procedure that leads to
$\widehat{m}_{ni}$. þ4.1 requires that $\widehat{m}$ be $L_{2}$ consistent for
the true conditional expectation $m(\cdot;Q)$ when the data are generated
according to $Q$ which belongs to an appropriately chosen subset
$\mathbf{Q}\left(\widehat{m}\right)\subseteq\mathbf{Q}$ which also contains
$Q_{0}$. Note that the $L_{2}$ convergence rate of $\widehat{m}$ can be
arbitrarily slow. For our purposes, simple consistency of this kind is enough.
However, $\mathbf{Q}\left(\widehat{m}\right)$ can be formed by restrictions
that impose a rate condition on $\widehat{m}$. For instance, the experimenter
might impose smoothness conditions required in existing literature for the
particular $\widehat{m}$ to obtain particular rates - e.g. a Hölder or Sobolev
ball. If $\widehat{m}$ is a series estimator, these can be found for example
in [43] and [6]. For nonlinear sieves such as neural networks, one can see for
[5], [13], [55] and [22]. For kernel estimators with uniform rates, see [39]
and [30]. Other restrictions may be functional form restrictions, e.g., random
forests are known to be $L_{2}$ consistent for additive models - see [56].
When $\mathbf{Q}\left(\widehat{m}\right)=\mathbf{Q}$, we will say that
$\widehat{m}$ is _universally $L_{2}$ consistent_. This is a property enjoyed
by the Nadaraya-Watson Kernel estimator - see for instance [19] and [57].
Next, we state assumptions that describe the cross-fitting approach to be used
here.

######  Assumption 4.2.

þ Let $J\in\mathbb{N}\setminus\\{1\\}$ be given. Let
$\mathbf{U}_{n}=\left\\{U_{n}(a,s):s\in\mathbb{N}_{\mathcal{S}},a\in\\{0,1\\}\right\\}$
be a set of independent $\mathrm{Uniform}([0,1])$ random variables on
$(\Omega,\mathcal{F},\Pr)$ such that
$\mathbf{U}_{n}\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}\left(\mathbf{W}_{n},\mathbf{A}_{n}\right)$.
Then,

$\left\\{\mathcal{G}_{n,j}(a,s):j\in\mathbb{N}_{J},a\in\\{0,1\\},s\in\mathbb{N}_{\mathcal{S}}\right\\}$

are random subsets of $\mathbb{N}_{n}$ satisfying the following for each
$s\in\mathbb{N}_{s}$ and $a\in\\{0,1\\}$.

  1. 1.

$\mathcal{G}_{n1}(a,s),\dots,\mathcal{G}_{nJ}(a,s)$ depend only on
$U_{n}(a,s)$ and $N_{n}(a,s)$.333Formally,
$\mathcal{G}_{n1}(a,s),\dots,\mathcal{G}_{nJ}(a,s)$ are measureable with
respect to the $\sigma$-algebra generated by the pair $U_{n}(a,s),N_{n}(a,s)$

  2. 2.

For each $s\in\mathbb{N}_{\mathcal{S}}$ and $a\in\\{0,1\\}$,
$\mathcal{G}_{n1}(a,s),\dots,\mathcal{G}_{nJ}(a,s)$ forms a partition of
$\left\\{i\in\mathbb{N}_{n}:A_{ni}=a,S_{i}=s\right\\}$ satisfying

$\begin{split}\left|\mathcal{G}_{n}(a,s,j)\right|=&\ \left\lfloor
N_{n}(a,s)/J\right\rfloor\quad j=1,\dots,J-1.\\\
\left|\mathcal{G}_{n}(a,s,J)\right|=&\ N_{n}(a,s)-(J-1)\cdot\left\lfloor
N_{n}(a,s)/J\right\rfloor.\end{split}$ (4.5)

For each $s\in\mathbb{N}_{\mathcal{S}}$ and $j\in\mathbb{N}_{J}$, define the
$j$th fold within stratum $s$ by

$\mathcal{G}_{n}(s,j)=\mathcal{G}_{n}(1,s,j)\cup\mathcal{G}_{n}(0,s,j).$ (4.6)

######  Assumption 4.3.

þ Let
$\left\\{\mathcal{G}_{nj}(s):j\in\mathbb{N}_{J},s\in\mathbb{N}_{\mathcal{S}}\right\\}$
be as in (4.6) in þ4.2 and let $\widehat{m}$ be as in þ4.1. For each
$i\in\mathbb{N}_{n}$, denote

$\Gamma_{n}(a,i)=\bigcup\left\\{\mathcal{G}_{nj}\left(a,S_{i}\right):j\in\mathbb{N}_{J}\text{
such that }i\notin\mathcal{G}_{nj}\left(S_{i}\right)\right\\}.$ (4.7)

The estimator $\widehat{m}_{ni}$ is defined by

$\widehat{m}_{ni}(a,z)=\widehat{m}\left(\left|\Gamma_{n}(a,i)\right|,a,z;\left\\{Y_{\iota}(a),Z_{\iota}:\iota\in\Gamma_{n}(a,i)\right\\}\right).$
(4.8)

######  Assumption 4.4.

þ $\widehat{\pi}_{n}(s)$ is either $\pi(s)$ or $N_{n}(s)^{-1}N_{n}(1,s)$.
Furthermore, if the latter is true, under $Q_{0}$ defined in þ2.1 and
$\alpha_{n}(\cdot|\cdot)$ in þ2.4, for each $s\in\mathbb{N}_{\mathcal{S}}$,

$\sqrt{n}\left|\frac{N_{n}(1,s)}{N_{n}(s)}-\pi(s)\right|=O_{\mathrm{p}}(1).$
(4.9)

þ4.2 first requires the experimenter to split each treatment group within a
given stratum into $J$ folds in a manner that is independent to the overall
sample using information only about the group size. These folds are further
required to satisfy an “equal size” requirement specific to the treatment
group within the stratum. For a given stratum, the two treatment groups
corresponding to fold $j$ are then combined which produces $J$ folds for the
overall stratum. We first split treatment groups by strata into folds to
ensure that a given combined fold has units in both treatment and control
groups. þ4.3 requires that the estimator $\widehat{m}_{ni}$ for $i$th
observation be constructed from $\widehat{m}$ in þ4.1 using observations in
the same stratum as $i$ but not in the same fold as $i$. Finally, þ4.4
requires that the treatment proportion estimator $\widehat{\pi}_{n}(s)$ will
be either the true treatment proportions or the sample treatment proportions.
For the latter, þ4.4 requires require consistency at a $1/\sqrt{n}$ rate. This
corresponds to requiring that $\alpha_{n}(\cdot|\cdot)$ in þ2.4 at least
satisfy “weak balance” as defined in the paragraph after þ2.2. The key result
is þ4.1 below.

######  Theorem 4.1.

þ Let þLABEL:asm--Q,asm--iid-Q,asm--treat-strat hold. Then

$\sqrt{n}\left(\widetilde{\beta}^{\ast}_{n}-\beta_{0}\right)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\varphi_{0}\left(Y_{ni},A_{ni},Z_{i}\right)\overset{\mathrm{d}}{\to}\mathcal{N}\left(0,\mathbb{V}_{\ast}\right).$
(4.10)

Suppose in addition that þLABEL:asm--mhat-L2,asm--sample-split,asm--
split-m,asm--weak-balance hold. Then $\widehat{\beta}^{\ast}_{n}$ in (4.3)
satisfies

$\sqrt{n}\left(\widehat{\beta}^{\ast}_{n}-\beta_{0}\right)=\sqrt{n}\left(\widetilde{\beta}^{\ast}_{n}-\beta_{0}\right)+o_{p}(1)$
(4.11)

so that $\widehat{\beta}^{\ast}_{n}$ achieves the semiparametric efficiency
bound $\mathbb{V}_{\ast}$.

þ4.1 first establishes in (4.10) that the infeasible ideal estimator
$\widetilde{\beta}^{\ast}_{n}$ achieves the efficiency bound in (3.1) under
general CAR procedures. This implies that any estimator of the ATE that is
asymptotically linear with influence function $\varphi_{0}$ in (3.4) achieves
the semiparametric efficiency bound. þ4.1 then establishes in (4.11) that
$\widehat{\beta}^{\ast}_{n}$ as defined by (4.3) has this aforementioned
property (under the additional assumption of this section). The only
additional restrictions placed on the overall semiparametric model
$\mathbf{Q}$ are ones defining $\mathbf{Q}\left(\widehat{m}\right)$ specific
to the choice of estimator $\widehat{m}$ in þ4.1. As a further corollary (þ4.2
below), we can show that $\mathbf{Q}\left(\widehat{m}\right)=\mathbf{Q}$ when
$\widehat{m}$ is the Nadaraya-Watson kernel regression estimator. Note that
the significance of $\mathbf{Q}\left(\widehat{m}\right)=\mathbf{Q}$ is that
efficient estimation is possible (pointwise) over the whole of $\mathbf{Q}$.
That is, the efficiency bound can be achieved under the same conditions used
for its derivation. This is a novel result since the overall semiparametric
problem is non-trivial and involves estimation of (possibly) infinite-
dimensional conditional means. We now first describe the Nadaraya-Watson
estimator, state assumptions on its tuning parameters and then provide the
result. In what follows, we use the normalization that $0/0=0$. Let
$\widehat{m}$ in þ4.1 be defined by

$\widehat{m}\left(n,a,z;y_{1}(a),z_{1},\dots,Y_{n}(a),z_{n}\right)=\frac{\sum_{j=1}^{n}y_{j}\cdot\kappa\left(h_{n}^{-1}\left(z_{j}-z\right)\right)}{\sum_{j=1}^{n}\kappa\left(h_{n}^{-1}\left(z_{j}-z\right)\right)}.$
(4.12)

þ4.5 below states the assumptions required for the bandwidth sequence
$\left\\{h_{n}:n\in\mathbb{N}\right\\}$ and the kernel function $\kappa$.

######  Assumption 4.5.

þ The sequence $\left\\{h_{n}:n\in\mathbb{N}\right\\}$ and the function
$\kappa:\mathbb{R}^{k}\to\mathbb{R}$ satisfy the following.

  1. (a)

$h_{n}>0$ for each $n\in\mathbb{N}$, $\lim_{n\to\infty}h_{n}=0$ and
$\lim_{n\to\infty}nh_{n}^{k}=\infty$.

  2. (b)

For some $\underline{\kappa},\overline{\kappa},r,R\in(0,\infty)$,
$\underline{\kappa}\cdot\mathbb{I}(\|u\|\leq
r)\leq\kappa(u)\leq\overline{\kappa}\cdot\mathbb{I}(\|u\|\leq R)$ for each
$u\in\mathbb{R}^{k}$.

þ4.5 maintains exactly the same conditions required by [19] and [57] for
universal $L_{2}$ consistency. Note that þ4.5 (b) requires the kernel function
$\kappa$ to be non-negative, to have compact support and to be strictly
positive in a closed neighborhood of the origin with non-empty interior. The
following result shows that efficient estimation is possible under exactly the
same conditions required to derive the efficiency bound.

######  Corollary 4.2.

þ In addition to the conditions of þ4.1, suppose that $\widehat{m}$ in þ4.1 is
defined by (4.12) and that $\left\\{h_{n}:n\in\mathbb{N}\right\\}$ and
$\kappa$ satisfy þ4.5. Then $\mathbf{Q}\left(\widehat{m}\right)=\mathbf{Q}$
and $\widehat{\beta}^{\ast}_{n}$ in (4.11) achieves the efficiency bound over
all of $\mathbf{Q}$.

The only requirement for the phenomenon in þ4.2 to occur is the universal
consistency requirement, i.e. that
$\mathbf{Q}\left(\widehat{m}\right)=\mathbf{Q}$. This is not special to the
Nadaraya-Watson estimator - one can replace this with a nearest neighbor
estimator and replace the conditions in þ4.5 with those of [59]. As for local
polynomial kernel estimators, these are not universally $L_{2}$ consistent in
their typical form (see [25, Section 5.4, p. 80-81 ]), but adjusted versions
can have the universal $L_{2}$ consistency property (see [34]). These
adjustments introduce a number of additional tuning parameters beyond the
bandwidth. For universal $L_{2}$ consistency conditions for series estimators,
see [25, Chapter 10 ].

We now provide a brief sketch of the main arguments that lead to (4.11) in
þ4.1. The difference between
$\sqrt{n}\left(\widehat{\beta}_{n}^{\ast}-\beta_{0}\right)$ and
$\sqrt{n}\left(\widetilde{\beta}_{n}^{\ast}-\beta_{0}\right)$ and can be
summarized as

$\begin{split}\sqrt{n}\left(\widehat{\beta}^{\ast}_{n}-\beta_{0}\right)=&\
\sqrt{n}\left(\widetilde{\beta}^{\ast}_{n}-\beta_{0}\right)+R_{1,n}-R_{0,n}+\widetilde{R}_{1n}-\widetilde{R}_{0n}\\\
\text{where}\quad R_{a,n}=&\
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbb{I}\left\\{A_{ni}=a\right\\}-\widehat{\pi}_{n,a}\left(\mathbb{S}\left(Z_{i}\right)\right)}{\widehat{\pi}_{n,a}\left(\mathbb{S}\left(Z_{i}\right)\right)}\left[\widehat{m}_{ni}\left(a,Z_{i}\right)-m_{\ast}\left(a,Z_{i}\right)\right].\\\
\text{and}\quad\widetilde{R}_{a,n}=&\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[\frac{1}{\widehat{\pi}_{n,a}\left(\mathbb{S}\left(Z_{i}\right)\right)}-\frac{1}{\pi_{a}\left(\mathbb{S}\left(Z_{i}\right)\right)}\right]\mathbb{I}\left\\{A_{ni}=a\right\\}\left[Y_{ni}-m_{\ast}\left(a,Z_{i}\right)\right].\end{split}$
(4.13)

In the above, $\pi_{a}(s)=\pi(s)^{a}[1-\pi(s)]^{1-a}$ and
$\widehat{\pi}_{n,a}(s)$ is defined analogously for each
$s\in\mathbb{N}_{\mathcal{S}}$ and $a\in\\{0,1\\}$. Showing (4.11) is a matter
of showing that $R_{a,n}=o_{\mathrm{p}}(1)$ and
$\widetilde{R}_{a,n}=o_{\mathrm{p}}(1)$. The remainder term
$\widetilde{R}_{a,n}$ is straightforward to deal with - a central limit
theorem applies to

$(1/\sqrt{n})\sum_{i=1}^{n}\mathbb{I}\left\\{A_{ni}=a,\mathbb{S}\left(Z_{i}\right)=s\right\\}\left[Y_{ni}-m_{\ast}\left(a,Z_{i}\right)\right]$

and the difference between estimated and true inverse treatment probabilities
is constant across observations within a given stratum and converging to zero.
For $R_{a,n}$, we deal with these by first splitting them into a sum of
$\mathcal{S}\cdot J$ terms each corresponding to a given stratum and a given
fold within that stratum. Each of these terms can be treated separately since
we are trying to prove convergence in probability to zero. The argument then
proceeds by noting that within the stratum-fold remainder each component of
the form
$\widehat{\pi}_{n,a}(s)^{-1}\left(\mathbb{I}\left\\{A_{ni}=a\right\\}-\widehat{\pi}_{n,a}(s)\right)$
is “approximately” mean zero and has finite variance. Furthermore, for the
term $\widehat{m}_{ni}\left(a,Z_{i}\right)-m_{\ast}\left(a,Z_{i}\right)$ has
second moment tending to zero. By careful use of the independence between the
evaluation point $Z_{i}$ and the estimation sample (observations outside the
corresponding fold), the stratum-fold specific version of $R_{a,n}$ is shown
to be $o_{\mathrm{p}}(1)$ via Chebychev’s inequality. Since the overall
$R_{a,n}$ is the sum of its stratum-fold specific counterparts and there are
$\mathcal{S}\cdot J$ (i.e. finitely many) of these, the conclusion follows
from Slutsky’s theorem. A rigorous version of these arguments is left to the
appendix.

The key finding here is that knowledge of the propensity score and its finite
support structure reduces the burden for efficient estimation considerably.
The fact that knowledge of the propensity score can reduce this burden was
also noticed by [2] and [52]. [2] show that potentially non-linear parametric
regression adjustments can improve estimation accuracy, generalizing [66] and
[37] to the nonlinear case. [52] shows that semiparametrically efficient
estimation is possible under a fairly broad class of estimators using Donsker
conditions. [52] also shows that efficient estimation is possible under
slightly weaker conditions by using a leave-one-out locally linear kernel
regression estimator by leveraging the results of [53]. However, both still
require dimension-dependent smoothness conditions on the conditional means to
be estimated. To the best of our knowledge, existing results using
nonparametric estimators in this context all require existence of densities
for baseline covariates that are bounded away from zero on their support and
higher order differentiability on conditional means and possibly also
densities. Our results do not require any of these for efficient estimation.

It should be noted that while $\widehat{\beta}^{\ast}_{n}$ in (4.3) achieves
the efficiency bound a few issues prevent it from being usable immediately in
the context of statistical inference. In particular, we have not provided
consistent estimators of the asymptotic variance $\mathbb{V}_{\ast}$ in (3.1).
It may be possible to construct these directly from the sample and the
nonparametric estimators $\widehat{m}_{ni}$. Another possibility might be a
bootstrap procedure to produce confidence intervals for hypothesis tests.
Another issue we have not addressed is data-based choice of tuning parameters
for the estimators $\widehat{m}_{ni}$ when these are based on the Nadaraya-
Watson kernel estimator. Since our main questions are the efficiency bound and
the conditions under which it can be achieved, we leave the construction of
consistent inference procedures from $\widehat{\beta}^{\ast}_{n}$ and data-
based tuning parameter selection for future research.

##  5 Monte Carlo evidence

In this section, we examine the finite sample performance of feasible
efficient estimator $\widehat{\beta}_{n}^{\ast}$ from (4.3). We start by
describing the data generating processes (DGPs) for our simulations. We
consider values of $n\in\\{500,1000,2000,4000,8000\\}$ and $k\in\\{1,5\\}$.
The DGPs are all of the form

$Y_{i}(a)=m_{\ast}\left(a,Z_{i}\right)+\sigma\left(a,Z_{i}\right)\cdot\varepsilon_{i}.$
(5.1)

In all cases, we have $Z_{i}\sim\mathrm{Uniform}\left([-1,1]^{k}\right)$ and
$\varepsilon_{i}\sim\mathcal{N}(0,1)$. For each DGP, we conduct 5000 Monte
Carlo simulations. In each simulation round, we compute
$\widetilde{\beta}_{n}^{\ast}$ in (4.2) and $\widehat{\beta}_{n}^{\ast}$ in
(4.3) as well as additional estimators of the ATE to be described below. We
report mean squared errors as well as biases of these estimators relative to
the true value of the ATE. For stratification, we stratify on the basis of the
first component of the covariates $Z_{i}$. We consider
$\mathcal{S}\in\\{5,20\\}$ and for each value of $\mathcal{S}$, strata are
constructed by splitting the interval $[-1,1]$ into adjacent segments of
length $2/\mathcal{S}$. We report results with constant assignment proportions
across all strata as well as proportions that vary by strata. We present
results with SPBR here, and defer results with SSRA to the appendix.
Inspecting both, the reader can see that the choice of assignment mechanism
has no visible impact on the performance of any of the estimators considered
here. For constant proportions, we set assignment proportions to 1/2 and for
varying proportions we use

$\bm{\pi}=\begin{cases}(0.3,0.4,0.5,0.6,0.7)&\text{if }\mathcal{S}=5,\\\
(0.325,0.35,0.375,\dots,0.75,0.775,0.8)&\text{if }\mathcal{S}=20.\end{cases}$
(5.2)

We use the boldfaced $\bm{\pi}$ here to distinguish assignment probabilities
from the mathematical constant $\pi=3.14159\dots$, since our choice of
conditional mean functions involve the latter through trigonometric functions.
We consider four different DGP’s by choice of $k$, $m_{\ast}(a,\cdot)$ and
$\sigma(a,\cdot)$. In what follows, DGP’s 1 and 2 have $k=1$ whereas DGP’s 3
and 4 have $k=5$. In all DGP’s, the true value of the ATE is $\beta_{0}=0$.
This is done for convenience.

  1. 1.

DGP 1: $m_{\ast}(0,z)=\sin(10\pi z)$, $m_{\ast}(1,z)=m_{\ast}(0,z)+2\cos(10\pi
z)$, $\sigma(0,z)=1+|z|$ and $\sigma(1,z)=\sqrt{2}\cdot\sigma(0,z)$

  2. 2.

DGP 2: $m_{\ast}(0,z)=\mathrm{sign}(z)\cdot\lfloor 10z\rfloor/10$,
$m_{\ast}(1,z)=m_{\ast}(0,z)+2m_{\ast}(0,z)^{3}$, $\sigma(\cdot)$ as in DGP 1.

  3. 3.

DGP 3: $\sigma(0,z)=1$, $\sigma(1,z)=\sqrt{2}$,
$m_{\ast}(0,z)=\lambda\left(z_{5},z_{4},z_{3},z_{2},z_{1}\right)$ and
$m_{\ast}(1,z)=\lambda(z)+2\sin\left(2\pi z_{1}z_{2}\right)$

$\lambda(z)=\cos\left(2\pi
z_{1}z_{2}\right)+\left(z_{3}+z_{4}-1\right)^{2}+\frac{z_{5}}{2}$

  4. 4.

DGP 4: $\sigma(\cdot)$ and $m_{\ast}(\cdot)$ as in DGP 3, but with $\lambda$
defined instead by

$\lambda(z)=\ \cos\left(2\pi
z_{1}z_{2}\right)+\left(z_{3}+z_{4}-1\right)^{2}+\frac{z_{5}}{2}+\mathbb{I}\left(z_{1}\geq
0\right).$

In both DGP 1 and DGP 3, $m_{\ast}(1,\cdot)$ and $m_{\ast}(0,\cdot)$ are
analytic and exhibit considerable non-linear variation within strata with
either $\mathcal{S}=5$ or $\mathcal{S}=20$. In DGP 2, $m_{\ast}(1,\cdot)$ and
$m_{\ast}(0,\cdot)$ have 19 discontinuities and are exactly fully saturated
regressions with $\mathcal{S}=20$ but not $\mathcal{S}=5$. DGP 4 adds a jump
discontinuity at $z_{1}=0$ to the conditional mean functions of DGP 3.

For comparison, we also examine the finite sample performance of the
infeasible efficient estimator $\widetilde{\beta}_{n}^{\ast}$, the fully
saturated regression estimator of the ATE $\widehat{\beta}_{n,\mathrm{SAT}}$
in (3.2), as well as the imputation estimator of [26] using the cross-fitted
Nadaraya-Watson estimator:

$\widehat{\beta}_{n,\mathrm{IMP}}=\frac{1}{n}\sum_{i=1}^{n}\left[Y_{ni}A_{ni}+\left(1-A_{ni}\right)\widehat{m}_{n,i}\left(1,Z_{i}\right)\right]-\frac{1}{n}\sum_{i=1}^{n}\left[\left(1-A_{ni}\right)Y_{ni}+A_{ni}\widehat{m}_{n,i}\left(0,Z_{i}\right)\right].$
(5.3)

The estimator above is an imputation estimator since it imputes the treatment
$a$ potential outcome for observation $i$ with the predicted value
$\widehat{m}_{ni}\left(a,Z_{i}\right)$ whenever that potential outcome is
unobserved. This estimator is also asymptotically linear and with influence
function $\varphi_{0}$ when the functions $m_{\ast}(a,\cdot)$ are sufficiently
smooth - see for instance [14]. For our nonparametric estimators
$\widehat{m}_{n,i}\left(a,\cdot\right)$, we use bandwidths of the form
$c_{k}n^{-1/(4+k)}$ and the uniform kernel so that þ4.5 is satisfied. For DGP
1 and DGP 3, the bandwidths are rate-optimal in terms of integrated mean-
squared error (see [60]) since the conditional mean functions
$m_{\ast}(a,\cdot)$ are twice continuously differentiable. For $k=1$, we set
$c_{k}=1/\sqrt{3}$ and for $k=5$, we set $c_{k}=3$. In our results, estimated
mean squared errors for $\widetilde{\beta}_{n}^{\ast}$ act as an estimate of
the semiparametric efficiency bound. The comparison between
$\widehat{\beta}_{n}^{\ast}$ and $\widetilde{\beta}_{n}^{\ast}$ illustrates
how far from optimal the feasible estimator can be due to noise in the
estimation of the conditional mean functions. Following the discussion in
Section 3, the fully saturated regression estimator
$\widehat{\beta}_{n,\mathrm{SAT}}$ in (3.2) achieves the semiparametric
efficiency bound among all estimators that use only information contained in
the strata. This is $\mathbb{V}_{\mathrm{SAT}}$, defined in (3.3). Thus,
comparing $\widehat{\beta}_{n,\mathrm{SAT}}$ to $\widetilde{\beta}_{n}^{\ast}$
illustrates the magnitude of efficiency loss from ignoring information from
the covariates (except in DGP 2 with $\mathcal{S}=20$). Additionally,
comparing $\widehat{\beta}_{n,\mathrm{SAT}}$ to $\widehat{\beta}_{n}^{\ast}$
illustrates the possible tradeoff between finite sample and asymptotic
accuracy when estimates of the nonparametric components are potentially far
from the truth. The additional nonparametric estimator
$\widehat{\beta}_{n,\mathrm{IMP}}$ has the same influence function as
$\widehat{\beta}_{n}^{\ast}$ but does not directly use the structure of the
influence function. Thus, comparisons between these estimators illustrates the
debiasing effect of using the efficient influence function directly.

Tables 1 and 2 present results from simulations with constant target treatment
proportions across strata with $\mathcal{S}=5$ and $\mathcal{S}=20$
respectively. Tables 3 and 4 present results with varying target treatment
proportions across strata with $\mathcal{S}=5$ and $\mathcal{S}=20$
respectively. Some common patterns appear across examples. In the univariate
case (DGP’s 1 and 2), $\widehat{\beta}^{\ast}_{n}$ is fairly close to the
infeasible estimator $\widetilde{\beta}^{\ast}_{n}$ in terms of mean squared
error. Furthermore, while $\widehat{\beta}_{n,\mathrm{SAT}}$ is far from
optimal under the nonlinearities in DGP 1 and $\mathcal{S}=5$, the distance is
greatly reduced by using finer strata (i.e. $\mathcal{S}=20$). However, in
higher dimensions, both estimators suffer albeit in different ways. Under
DGP’s 3 and 4 the feasible optimal estimator $\widehat{\beta}^{\ast}_{n}$,
despite still being consistent and asymptotically unbiased, is much slower in
its tendency towards the SPEB. This is mainly due to the curse of
dimensionality. In simulations with $n\in\\{10000,20000\\}$ for instance,
$\widehat{\beta}^{\ast}_{n}$ is closer to achieving the SPEB, but these sample
sizes may be unrealistic for most experiments. As we can see,
$\widehat{\beta}_{n}^{\ast}$ is also sensitive to the number of strata -
having more strata reduces the effective estimation sample used for estimation
of the conditional mean. This is particularly stark in the examples with
varying treatment proportions - in these, some treatment groups are going to
have small sample sizes within strata by design and the aforementioned
phenomenon is exacerbated. In the multivariate case, increasing the number of
strata does not help reduce the MSE of $\widehat{\beta}_{n,\mathrm{SAT}}$
since stratification is done on the basis of a single component of a
continuously distributed covariate. On the other hand, though its performance
in some instances (e.g. 4, DGP 3 and 4) leaves much to be desired, in most
cases with DGP 3 and 4, $\widehat{\beta}^{\ast}_{n}$ picks up on the nonlinear
variation in the conditional means across all dimensions. To summarize the
comparison between $\widehat{\beta}^{\ast}_{n}$ and
$\widehat{\beta}_{n,\mathrm{SAT}}$ we find that across all simulation designs,
$\widehat{\beta}^{\ast}_{n}$ is on average 13% more efficient (averaging the
quotient of the MSEs). The maximal gain from using
$\widehat{\beta}^{\ast}_{n}$ is a 40% reduction in MSE - Table 3, DGP 4 with
$n=8000$.

For $\widehat{\beta}_{n,\mathrm{IMP}}$, while it seems to perform well under
$k=1$ and $\mathcal{S}=20$ the impact of the curse of dimensionality and its
interaction with the lack of debiasing is quite stark as can be seen from the
other cases. Furthermore, the need for undersmoothing to reduce bias for this
estimator is apparent since it seems to sometimes do worse with more effective
observations. For instance, comparing results for DGP 1 in Tables 1 and 2, we
see that increasing the number of strata (i.e. decreasing the effective number
of observations per strata) produces bias reductions without seriously
affecting variance for $\widehat{\beta}_{n,\mathrm{SAT}}$ or
$\widehat{\beta}_{n,\mathrm{IMP}}$. However, reducing bandwidth to undersmooth
will reduce bias and inflate the variance of this estimator.

Table 1: $\mathcal{S}=5$ and constant target assignments across strata. |  | $\sqrt{n}\cdot\widetilde{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{SAT}}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{IMP}}$
---|---|---|---|---|---|---|---|---
DGP | $n$ | MSE | Bias |  | MSE | Bias |  | MSE | Bias |  | MSE | Bias
DGP 1 | 500 | 16.442 | -0.041 |  | 19.883 | -0.078 |  | 20.931 | -0.071 |  | 22.536 | -1.471
1000 | 15.731 | 0.024 |  | 17.724 | -0.014 |  | 19.589 | -0.040 |  | 21.316 | -1.661
2000 | 16.380 | -0.064 |  | 17.888 | -0.087 |  | 20.631 | -0.082 |  | 22.458 | -1.908
4000 | 16.084 | -0.039 |  | 17.022 | -0.076 |  | 19.891 | -0.085 |  | 21.825 | -2.011
8000 | 15.749 | 0.025 |  | 16.466 | 0.033 |  | 20.167 | 0.058 |  | 20.990 | -1.955
DGP 2 | 500 | 14.679 | -0.012 |  | 15.139 | -0.020 |  | 14.928 | -0.014 |  | 15.090 | -0.016
1000 | 14.000 | -0.012 |  | 14.274 | -0.009 |  | 14.235 | -0.005 |  | 14.266 | -0.010
2000 | 14.604 | -0.058 |  | 14.733 | -0.061 |  | 14.836 | -0.067 |  | 14.749 | -0.060
4000 | 14.672 | -0.077 |  | 14.786 | -0.087 |  | 14.912 | -0.078 |  | 14.802 | -0.085
8000 | 14.296 | 0.006 |  | 14.327 | 0.006 |  | 14.511 | 0.000 |  | 14.326 | 0.005
DGP 3 | 500 | 12.593 | -0.011 |  | 18.453 | -0.016 |  | 24.129 | -0.078 |  | 18.291 | -0.685
1000 | 13.148 | -0.034 |  | 17.397 | -0.007 |  | 24.508 | -0.070 |  | 18.098 | -0.792
2000 | 12.974 | -0.019 |  | 16.380 | -0.036 |  | 24.091 | -0.008 |  | 17.481 | -0.912
4000 | 12.813 | -0.044 |  | 15.290 | -0.047 |  | 23.656 | -0.005 |  | 16.836 | -1.015
8000 | 13.060 | -0.014 |  | 15.155 | -0.030 |  | 24.447 | -0.107 |  | 16.987 | -1.087
DGP 4 | 500 | 12.624 | -0.021 |  | 18.743 | -0.030 |  | 24.668 | -0.090 |  | 18.566 | -0.702
1000 | 13.120 | -0.040 |  | 17.667 | -0.012 |  | 25.168 | -0.070 |  | 18.351 | -0.795
2000 | 12.932 | 0.001 |  | 16.449 | -0.014 |  | 24.615 | 0.011 |  | 17.523 | -0.893
4000 | 12.762 | -0.037 |  | 15.431 | -0.034 |  | 24.227 | 0.018 |  | 16.968 | -1.000
8000 | 13.006 | -0.020 |  | 15.197 | -0.031 |  | 25.097 | -0.109 |  | 17.076 | -1.090
Table 2: $\mathcal{S}=20$ and constant target assignments across strata. |  | $\sqrt{n}\cdot\widetilde{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{SAT}}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{IMP}}$
---|---|---|---|---|---|---|---|---
DGP | $n$ | MSE | Bias |  | MSE | Bias |  | MSE | Bias |  | MSE | Bias
DGP 1 | 500 | 16.290 | -0.125 |  | 18.864 | -0.126 |  | 19.008 | -0.128 |  | 18.919 | -0.133
1000 | 16.171 | -0.077 |  | 17.921 | -0.060 |  | 18.268 | -0.058 |  | 18.042 | -0.064
2000 | 16.582 | -0.151 |  | 17.939 | -0.146 |  | 18.680 | -0.143 |  | 18.225 | -0.149
4000 | 15.854 | -0.015 |  | 17.104 | 0.002 |  | 18.390 | 0.002 |  | 17.512 | 0.001
8000 | 15.462 | -0.014 |  | 16.265 | -0.024 |  | 17.811 | -0.020 |  | 16.688 | -0.021
DGP 2 | 500 | 14.549 | -0.102 |  | 14.785 | -0.100 |  | 14.744 | -0.100 |  | 14.769 | -0.108
1000 | 14.298 | -0.104 |  | 14.436 | -0.104 |  | 14.399 | -0.105 |  | 14.444 | -0.109
2000 | 14.878 | -0.161 |  | 14.966 | -0.158 |  | 14.933 | -0.162 |  | 15.039 | -0.161
4000 | 14.354 | -0.035 |  | 14.395 | -0.033 |  | 14.378 | -0.036 |  | 14.530 | -0.034
8000 | 13.917 | 0.000 |  | 13.954 | -0.001 |  | 13.928 | 0.000 |  | 14.091 | 0.001
DGP 3 | 500 | 13.114 | -0.039 |  | 24.858 | -0.640 |  | 24.637 | -0.038 |  | 18.169 | -1.151
1000 | 12.669 | -0.049 |  | 21.884 | -0.296 |  | 24.590 | -0.086 |  | 19.145 | -1.079
2000 | 12.894 | 0.024 |  | 19.287 | -0.114 |  | 24.830 | -0.003 |  | 18.681 | -1.070
4000 | 12.471 | -0.011 |  | 16.999 | -0.086 |  | 24.358 | -0.046 |  | 17.859 | -1.169
8000 | 13.402 | 0.013 |  | 16.501 | 0.004 |  | 24.877 | 0.070 |  | 18.007 | -1.088
DGP 4 | 500 | 13.127 | -0.049 |  | 26.421 | -0.797 |  | 25.091 | -0.056 |  | 19.007 | -1.313
1000 | 12.711 | -0.052 |  | 22.815 | -0.349 |  | 25.092 | -0.086 |  | 19.735 | -1.142
2000 | 12.893 | 0.041 |  | 19.856 | -0.110 |  | 25.412 | 0.020 |  | 18.990 | -1.073
4000 | 12.446 | 0.003 |  | 17.231 | -0.080 |  | 24.891 | -0.024 |  | 18.045 | -1.166
8000 | 13.396 | 0.019 |  | 16.616 | 0.013 |  | 25.588 | 0.077 |  | 18.091 | -1.079
Table 3: $\mathcal{S}=5$ and varying target assignments across strata. |  | $\sqrt{n}\cdot\widetilde{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{SAT}}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{IMP}}$
---|---|---|---|---|---|---|---|---
DGP | $n$ | MSE | Bias |  | MSE | Bias |  | MSE | Bias |  | MSE | Bias
DGP 1 | 500 | 18.224 | 0.050 |  | 22.458 | 0.031 |  | 22.951 | 0.059 |  | 24.469 | -1.342
1000 | 17.620 | -0.011 |  | 20.586 | -0.055 |  | 22.667 | -0.005 |  | 24.260 | -1.689
2000 | 17.951 | -0.036 |  | 19.725 | -0.040 |  | 22.685 | -0.034 |  | 24.258 | -1.866
4000 | 17.347 | -0.039 |  | 18.410 | -0.077 |  | 21.766 | -0.101 |  | 23.362 | -2.011
8000 | 17.519 | 0.029 |  | 18.221 | 0.003 |  | 22.196 | -0.027 |  | 23.027 | -1.996
DGP 2 | 500 | 16.578 | 0.035 |  | 17.596 | 0.003 |  | 16.942 | 0.047 |  | 17.289 | 0.019
1000 | 16.054 | -0.038 |  | 16.472 | -0.057 |  | 16.471 | -0.050 |  | 16.364 | -0.059
2000 | 16.054 | -0.031 |  | 16.248 | -0.021 |  | 16.359 | -0.027 |  | 16.245 | -0.025
4000 | 15.899 | -0.072 |  | 16.069 | -0.085 |  | 16.134 | -0.067 |  | 16.102 | -0.076
8000 | 16.179 | 0.046 |  | 16.208 | 0.049 |  | 16.379 | 0.048 |  | 16.244 | 0.049
DGP 3 | 500 | 13.967 | -0.004 |  | 21.786 | -0.046 |  | 27.781 | 0.023 |  | 20.290 | -0.527
1000 | 13.941 | 0.037 |  | 20.019 | 0.041 |  | 27.937 | 0.045 |  | 19.804 | -0.643
2000 | 13.738 | -0.049 |  | 18.115 | -0.097 |  | 27.328 | -0.098 |  | 19.272 | -0.955
4000 | 13.533 | -0.020 |  | 17.139 | -0.050 |  | 27.022 | -0.066 |  | 18.684 | -1.024
8000 | 13.858 | -0.023 |  | 16.250 | -0.001 |  | 26.544 | 0.013 |  | 18.027 | -1.028
DGP 4 | 500 | 14.112 | -0.011 |  | 22.398 | -0.059 |  | 28.679 | 0.019 |  | 20.594 | -0.392
1000 | 13.959 | 0.043 |  | 20.355 | 0.046 |  | 28.533 | 0.050 |  | 19.955 | -0.564
2000 | 13.741 | -0.037 |  | 18.338 | -0.079 |  | 27.870 | -0.081 |  | 19.413 | -0.913
4000 | 13.599 | -0.016 |  | 17.232 | -0.045 |  | 27.575 | -0.060 |  | 18.777 | -1.012
8000 | 13.847 | -0.025 |  | 16.363 | -0.002 |  | 27.180 | 0.024 |  | 18.175 | -1.029
Table 4: $\mathcal{S}=20$ and varying target assignments across strata. |  | $\sqrt{n}\cdot\widetilde{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}^{\ast}_{n}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{SAT}}$ |  | $\sqrt{n}\cdot\widehat{\beta}_{n,\mathrm{IMP}}$
---|---|---|---|---|---|---|---|---
DGP | $n$ | MSE | Bias |  | MSE | Bias |  | MSE | Bias |  | MSE | Bias
DGP 1 | 500 | 17.538 | -0.023 |  | 20.095 | -0.009 |  | 19.849 | -0.020 |  | 19.852 | -0.018
1000 | 17.204 | -0.124 |  | 19.118 | -0.129 |  | 19.446 | -0.135 |  | 19.259 | -0.112
2000 | 17.350 | -0.052 |  | 19.123 | -0.050 |  | 19.965 | -0.050 |  | 19.456 | 0.001
4000 | 17.194 | 0.062 |  | 18.189 | 0.052 |  | 19.246 | 0.043 |  | 18.695 | 0.142
8000 | 17.317 | 0.083 |  | 18.153 | 0.105 |  | 19.540 | 0.107 |  | 18.505 | 0.233
DGP 2 | 500 | 16.160 | -0.028 |  | 16.506 | -0.031 |  | 16.188 | -0.025 |  | 16.278 | -0.025
1000 | 15.856 | -0.149 |  | 15.956 | -0.145 |  | 15.874 | -0.149 |  | 16.040 | -0.153
2000 | 15.556 | -0.046 |  | 15.576 | -0.046 |  | 15.560 | -0.047 |  | 15.683 | -0.048
4000 | 15.868 | 0.020 |  | 15.916 | 0.022 |  | 15.869 | 0.020 |  | 16.205 | 0.020
8000 | 15.960 | 0.110 |  | 15.972 | 0.113 |  | 15.959 | 0.111 |  | 16.053 | 0.114
DGP 3 | 500 | 13.420 | -0.090 |  | 30.729 | -1.572 |  | 26.697 | -0.067 |  | 24.945 | 2.900
1000 | 13.443 | -0.104 |  | 27.120 | -0.862 |  | 26.943 | -0.080 |  | 30.147 | 3.458
2000 | 13.725 | 0.017 |  | 23.629 | -0.333 |  | 27.433 | -0.006 |  | 30.384 | 3.372
4000 | 13.581 | -0.092 |  | 20.450 | -0.160 |  | 27.035 | -0.097 |  | 23.979 | 2.354
8000 | 13.925 | -0.037 |  | 18.690 | -0.020 |  | 27.818 | 0.015 |  | 19.863 | 1.235
DGP 4 | 500 | 13.467 | -0.082 |  | 33.787 | -1.886 |  | 27.377 | -0.058 |  | 36.226 | 4.364
1000 | 13.486 | -0.122 |  | 28.984 | -1.029 |  | 27.681 | -0.104 |  | 43.866 | 5.022
2000 | 13.700 | 0.032 |  | 24.703 | -0.361 |  | 28.221 | 0.023 |  | 42.664 | 4.820
4000 | 13.649 | -0.076 |  | 21.122 | -0.157 |  | 27.870 | -0.084 |  | 30.253 | 3.392
8000 | 13.821 | -0.024 |  | 18.879 | -0.007 |  | 28.533 | 0.037 |  | 21.760 | 1.827

##  6 Conclusion

In this paper, we have characterized the maximal gains in efficiency from
using baseline covariates beyond strata in RCT’s covariate adaptive
randomization is used. To do this, we have established a semiparametric
efficiency bound under CAR for such experiments. This is the minimum
estimation variance achievable if one uses all the relevant information the
data have to offer about the parameter of interest (the ATE here). If baseline
covariates are used, we use the information they contain about the ATE fully
through differences in conditional means, averaged over the support of the
covariates. With continuous covariates, or more covariates present than used
in stratification, a strict improvement over fully saturated regression
estimates of the ATE is possible. The efficiency bound established is shown to
be achievable under the same conditions used for its derivation by using a
leave one out Nadaraya-Watson kernel regression estimator. Simulation evidence
presented shows that this estimator can indeed reach the efficiency bound in
large samples. However, such improvements are not guaranteed in finite
samples, and reaching the efficiency bound can be slow especially in the
presence of many covariates due to the curse of dimensionality.

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Appendix for“Efficient Semiparametric Estimation of Average Treatment Effects
Under Covariate Adaptive Randomization”
Ahnaf Rafi

######  Assumption A.

þ
$\mathbf{W}^{\ast}=\left\\{W_{i}(s):i\in\mathbb{N},s\in\mathbb{N}_{\mathcal{S}}\right\\}$
is a sequence of random vectors defined on $(\Omega,\mathcal{F},\Pr)$
independent across $(i,s)\in\mathbb{N}\times\mathbb{N}_{\mathcal{S}}$ such
that $W_{i}^{\prime}(s)=\left(Y_{i}(0,s),Y_{i}(1,s),Z_{i}^{\prime}(s)\right)$
has the same marginal distribution as $W|\mathbb{S}(Z)=s$. Furthermore,
$\mathbf{W}^{\ast}$ is independent to the collection
$\left\\{\mathbf{W}\right\\}\cup\left\\{\mathbf{A}_{n}:n\in\mathbb{N}\right\\}$.

##  Appendix A Proofs of theorems in paper

###  A.1 Proof of Theorem 3.1

###### Proof of þ3.1.

By þC.2, the ATE parameter, $\beta(\cdot)$ defined by (2.5) is pathwise
differentiable with efficient influence function $\varphi_{0}$ defined in
(3.4). This means that $\beta(\cdot)$ has a derivative operator
$\dot{\beta}(\cdot)$ such that for any element of the tangent space in þC.1,
$h\in\dot{\mathbf{P}}$,

$\dot{\beta}(h)=\mathbb{E}_{P_{0}}\left[\varphi_{0}(Y,A,Z)h(Y,A,Z)\right].$

The adjoint operator is the unique linear operator
$\dot{\beta}^{\ast}:\mathbb{R}\to L_{2}\left(P_{0}\right)$ defined by

$c\dot{\beta}(h)=c\mathbb{E}_{P_{0}}\left[\varphi_{0}(Y,A,Z)h(Y,A,Z)\right]=\mathbb{E}_{P_{0}}\left[\left\\{\dot{\beta}^{\ast}(c)\right\\}(Y,A,Z)h(Y,A,Z)\right].$

Clearly, $\left\\{\dot{\beta}^{\ast}(c)\right\\}(y,a,z)=c\varphi_{0}(y,a,z)$
for every $c\in\mathbb{R}$. For a given $h$, let $\mathbf{P}_{0h}$ be any one-
dimensional regular parametric submodel of $\mathbf{P}$ passing through
$P_{0}$ that has score $h$ at $P_{0}$. As in þ3.1 and þ3.1, this means that
$\mathbf{P}_{0h}=\left\\{P_{\theta,h}\in\mathbf{P}:\theta\in(-2,2)\right\\}$,
each $P_{\theta,h}$ has a density against $\nu$ in (3.5) of the form

$p(y,a,z;\theta,h)=\left[q_{1}(y,z;\theta,h)\pi(\mathbb{S}(z))\right]^{a}\left[q_{0}(y,z;\theta,h)\cdot\pi(1-\mathbb{S}(z))\right]^{1-a}.$

The requirement that $\mathbf{P}_{0h}$ passes through $P_{0}$ will be taken to
mean that $P_{\theta,h}=P_{0}$ implies $\theta=0$ (i.e. $P_{0}$ is identified
uniquely at $\theta=0$). We set $\theta_{0}=0$ since we can always use a
translation to ensure this without changing any conclusions. The choice
$\Theta=(-2,2)$ is to ensure $\Theta$ contains local alternatives of the form
$\theta_{n}=1/\sqrt{n}$ \- the conclusions are unaffected by redefining
$\Theta$ to be an open interval centered at $0$ and then re-scaling. By þ3.2,
regularity of $\mathbf{P}_{0h}$ means that the maps $\theta\mapsto
q_{a}(\cdot;\theta,h)$ has a derivative in quadratic mean, $D_{a,h}$ so that

$\lim_{\theta\to
0}\theta^{-2}\int\left(\sqrt{q_{a}(y,z;\theta,h)}-\sqrt{q_{a}\left(y,z;0,h\right)}-D_{a,h}(y,z)\cdot\theta\right)^{2}\nu_{a}(\mathrm{d}y,\mathrm{d}z)=0.$

The requirement that $h$ be the score function at $P_{0}$ corresponds to the
requirement that (by (3.10))

$h(y,a,z)=2\frac{D_{a,h}(y,z)}{\sqrt{q_{a}\left(y,z;Q_{0}\right)}}\mathbb{I}\left\\{q_{a}\left(y,z;Q_{0}\right)>0\right\\}.$

Let $\mathcal{P}_{h}=\left\\{\mathbf{P}_{nh}:n\in\mathbb{N}\right\\}$ be a
parametric submodel of $\mathcal{P}$ characterized by the maps $\theta\mapsto
q_{a}(\cdot;\theta,h)$ for $\theta\in(-2,2)$. That is, each element
$P_{n,\theta,h}\in\mathbf{P}_{nh}$ for $\theta\in(-2,2)$ has a density against
the $n$-fold product measure $\nu^{n}$ of the form

$\displaystyle
p_{n}\left(\mathbf{y}_{n},\mathbf{a}_{n},\mathbf{z}_{n};\theta,h\right)=$
$\displaystyle\
\alpha_{n}\left(\mathbf{a}_{n}|\mathbf{s}_{n}\right)\cdot\prod_{i=1}^{n}q_{1}\left(y_{i},z_{i};\theta,h\right)^{a_{i}}q_{0}\left(y_{i},z_{i};\theta,h\right)^{1-a_{i}},$
$\displaystyle\text{where }\mathbf{s}_{n}^{\prime}=$ $\displaystyle\
\left(\mathbb{S}\left(z_{1}\right),\dots,\mathbb{S}\left(z_{n}\right)\right)$

and $\alpha_{n}(\cdot|\cdot)$ satisfies þ2.4 (d). By þ3.2, every regular
parametric submodel of $\mathcal{P}$ has the LAN property, and hence
$\mathcal{P}_{h}$ necessarily exhibits the LAN property. In what follows, set
$\beta_{nh}=\beta\left(P_{\theta_{n},h}\right)$ for $\theta_{n}=1/\sqrt{n}$.
Note that a direct implication of þC.2 is that the local parameters
$\beta_{nh}$ are regular in the sense of [63, p. 413 ].

By Theorem 3.11.2 of [63, p. 414 ] if $\left\\{\widehat{\beta}_{n}\right\\}$
is a sequence of regular estimators for $\beta(\cdot)$, then
$\sqrt{n}\left(\widehat{\beta}_{n}-\beta_{0}\right)\overset{\mathrm{d}}{\to}G+W$
where $G$ and $W$ are independent random variables and

$G\sim\mathcal{N}\left(0,\mathbb{E}_{P_{0}}\left[\varphi_{0}(Y,A,Z)^{2}\right]\right)=\mathcal{N}\left(0,\mathbb{V}_{\ast}\right).$
(A.1)

This is the usual convolution theorem based justification of
$\mathbb{V}_{\ast}$ as a semiparametric efficiency bound. Next, let
$\mathcal{L}:\mathbb{R}\to\mathbb{R}$ be any bowl-shaped loss function, i.e.
$\mathcal{L}$ is non-negative, satisfies $\mathcal{L}(y)=\mathcal{L}(-y)$ and
has convex sublevel sets. By Theorem 3.11.5 of [63, p. 417 ], for any (not
necessarily regular) estimator sequence $\left\\{\widehat{\beta}_{n}\right\\}$
for $\beta(\cdot)$,

$\sup_{\begin{subarray}{c}I\subseteq\dot{\mathbf{P}}\\\ I\text{ is
finite}\end{subarray}}\liminf_{n\to\infty}\sup_{h\in
I}\mathbb{E}_{nh}\left[\mathcal{L}\left(\sqrt{n}\left(\widehat{\beta}_{n}-\beta_{nh}\right)\right)\right]\geq\mathbb{E}\left[\mathcal{L}(G)\right].$

Here, $G$ is as in (A.1) and $\mathbb{E}_{nh}$ denotes expectations taken with
respect to local perturbations of $P_{0,n}$ of the form $P_{n,\theta_{n},h}$
with $\theta_{n}=1/\sqrt{n}$. This is the justification of $\mathbb{V}_{\ast}$
as the semiparametric efficiency bound via the local asymptotic minimax
theorem. ∎

###  A.2 Proof of Theorem 4.1

For the purposes of the proof, it will be useful to define some additional
notation. For each observation $i\in\mathbb{N}_{n}$, let the corresponding
stratum-fold label in (4.6) (þ4.2) be

$\mathbb{J}_{i}=\sum_{s=1}^{\mathcal{S}}\sum_{j=1}^{J}j\cdot\mathbb{I}\left\\{\mathbb{S}\left(Z_{i}\right)=s,i\in\mathcal{G}_{nj}(s)\right\\}.$
(A.2)

Let

$\displaystyle\widehat{N}_{n}(a,s,j)=$ $\displaystyle\
\sum_{i=1}^{n}\mathbb{I}\left\\{A_{ni}=a,\mathbb{S}\left(Z_{i}\right)=s,\mathbb{J}_{i}=j\right\\},$
(A.3) $\displaystyle\widehat{N}_{n}(s,j)=$ $\displaystyle\
\sum_{i=1}^{n}\mathbb{I}\left\\{\mathbb{S}\left(Z_{i}\right)=s,\mathbb{J}_{i}=j\right\\}$
(A.4) $\displaystyle\widehat{N}_{n}^{\ast}(a,s,j)=$ $\displaystyle\
N_{n}(a,s)-\widehat{N}_{n}(a,s,j).$ (A.5)

$\widehat{N}(a,s,j)$ is the size of the $j$th fold for treatment group $a$
within stratum $s$. Similarly, $\widehat{N}_{n}(s,j)$ is the size of the $j$th
fold for stratum $s$. Finally, $\widehat{N}_{n}^{\ast}(a,s,j)$ in (A.5) is the
size of the estimation sample used for estimation of $m(a,\cdot)$ in fold $j$
of stratum $s$.

###### Proof of þ4.1.

In þ4.1, (4.10) follows from þC.6. To see that the conditions are satisfied,
first decompose $\varphi_{0}$ as

$\displaystyle\varphi_{0}(y,a,z)=$ $\displaystyle\
h(y,a,z)+\xi(\mathbb{S}(z))$ $\displaystyle h(y,a,z)=$ $\displaystyle\
\frac{a}{\pi(\mathbb{S}(z))}\left[y-m_{\ast}(1,z)\right]-\frac{1-a}{1-\pi(\mathbb{S}(z))}\left[y-m_{\ast}(0,z)\right]$
$\displaystyle+\left(m_{\ast}(1,z)-m_{\ast}(0,z)-\mathbb{E}\left[m_{\ast}(1,Z)-m_{\ast}(0,Z)|\mathbb{S}(Z)=\mathbb{S}(z)\right]\right)$
$\displaystyle\xi(s)=$ $\displaystyle\
\mathbb{E}\left[m_{\ast}(1,Z)-m_{\ast}(0,Z)|\mathbb{S}(Z)=s\right]-\beta_{0}.$

Note that $\mathbb{E}\left[(\varphi_{0}(Y(a),a,Z))^{2}\right]<\infty$ which
immediately implies that both $h$ and $\xi$ have finite second moments.
Furthermore, the (conditional and unconditional) mean zero requirements of
þC.6 are satisfied by the Law of Iterated Expectations. The fact that the
corresponding variance terms $\mathbb{V}_{h}$ and $\mathbb{V}_{\xi}$ from
(C.24) in þC.6 satisfy $\mathbb{V}_{h}+\mathbb{V}_{\xi}=\mathbb{V}_{\ast}$
follows from the Law of Total Variance:

$\displaystyle\mathbb{E}\left[\left(m_{\ast}(1,z)-m_{\ast}(0,z)-\beta_{0}\right)^{2}\right]=$
$\displaystyle\ \mathrm{Var}\left[m_{\ast}(1,z)-m_{\ast}(0,z)\right]$
$\displaystyle=$ $\displaystyle\
\mathbb{E}\left[\mathrm{Var}\left[m_{\ast}(1,z)-m_{\ast}(0,z)\middle|\mathbb{S}(Z)\right]\right]+\mathrm{Var}\left[\mathbb{E}\left[m_{\ast}(1,z)-m_{\ast}(0,z)\middle|\mathbb{S}(Z)\right]\right]$
$\displaystyle=$ $\displaystyle\
\mathbb{E}\left[\mathbb{E}\left[\left(m_{\ast}(1,z)-m_{\ast}(0,z)-\mathbb{E}\left[m_{\ast}(1,Z)-m_{\ast}(0,Z)\middle|\mathbb{S}(Z)\right]\right)^{2}\middle|\mathbb{S}(Z)\right]\right]$
$\displaystyle+\mathbb{E}\left[\left(\mathbb{E}\left[m_{\ast}(1,Z)-m_{\ast}(0,Z)|\mathbb{S}(Z)\right]-\beta_{0}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle\
\mathbb{E}\left[\left(m_{\ast}(1,z)-m_{\ast}(0,z)-\mathbb{E}\left[m_{\ast}(1,Z)-m_{\ast}(0,Z)\middle|\mathbb{S}(Z)\right]\right)^{2}\right]$
$\displaystyle+\mathbb{E}\left[\left(\mathbb{E}\left[m_{\ast}(1,Z)-m_{\ast}(0,Z)\middle|\mathbb{S}(Z)\right]-\beta_{0}\right)^{2}\right].$

Next, we show that (4.11) holds. By (4.13), it suffices to show that
$\widetilde{R}_{a,n}=o_{\mathrm{p}}(1)$ and $R_{a,n}=o_{\mathrm{p}}(1)$ for
each $a\in\\{0,1\\}$. We start with $\widetilde{R}_{a,n}$. Write

$\displaystyle\widetilde{R}_{a,n}=$
$\displaystyle\sum_{s=1}^{\mathcal{S}}\widetilde{R}_{n}(a,s),$
$\displaystyle\text{where}\quad\widetilde{R}_{n}(a,s)=$ $\displaystyle\
\left[\frac{1}{\widehat{\pi}_{n,a}(s)}-\frac{1}{\pi_{a}(s)}\right]\cdot\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\mathbb{I}\left\\{A_{ni}=a,\mathbb{S}\left(Z_{i}\right)=s\right\\}\left[Y_{ni}-m_{\ast}\left(a,Z_{i}\right)\right].$

Note that $Y_{i}(a)-m_{\ast}(a,Z_{i})$ has mean zero conditional on any given
stratum label (by the tower property of conditional expectations) and of
course has finite variance. Therefore, applying þC.6, we find immediately that
for any given $s\in\mathbb{N}_{\mathcal{S}}$,

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\mathbb{I}\left\\{A_{ni}=a,\mathbb{S}\left(Z_{i}\right)=s\right\\}\left[Y_{ni}-m_{\ast}\left(a,Z_{i}\right)\right]$

has a limit normal distribution. Then, for each $a\in\\{0,1\\}$ and
$s\in\mathbb{N}_{\mathcal{S}}$,

$\widetilde{R}_{n}(a,s)=\left[\frac{1}{\widehat{\pi}_{n,a}(s)}-\frac{1}{\pi_{a}(s)}\right]\cdot
O_{\mathrm{p}}(1)=o_{\mathrm{p}}(1)\cdot O_{\mathrm{p}}(1)=o_{\mathrm{p}}(1).$

Note that the first term is $o_{\mathrm{p}}(1)$ since $\widehat{\pi}_{n,a}(s)$
is either $\pi_{a}(s)$ (in which case this is trivially true), or
$N_{n}(s)^{-1}N_{n}(a,s)$ which is assumed to be consistent for
$\pi_{a}(s)\in(0,1)$. The conclusion follows for $\sqrt{n}\widetilde{R}_{a,n}$
using Slutsky’s Theorem.

Next, we show that $R_{a,n}=o_{\mathrm{p}}(1)$ for $R_{a,n}$ in (4.13) for any
$a\in\\{0,1\\}$. To do this, we will use the sample-splitting structure of the
estimators $\widehat{m}_{ni}$. Inspecting (4.13), write $R_{a,n}$ as

$\begin{split}R_{a,n}=&\ \sum_{s=1}^{\mathcal{S}}\sum_{j=1}^{J}R_{n}(a,s,j)\\\
\text{where}\quad R_{n}(a,s,j)=&\
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbb{I}\left\\{A_{ni}=a\right\\}-\widehat{\pi}_{n,a}(s)}{\widehat{\pi}_{n,a}(s)}\left[\widehat{m}_{ni}\left(a,Z_{i}\right)-m_{\ast}\left(a,Z_{i}\right)\right]\mathbb{I}\left\\{\mathbb{S}\left(Z_{i}\right)=s,\mathbb{J}_{i}=j\right\\}.\end{split}$
(A.6)

Since there are finitely many folds within finitely many strata, it is
sufficient to show that $R_{n}(a,s,j)=o_{\mathrm{p}}(1)$ for each
$s\in\mathbb{N}_{\mathcal{S}}$, $j\in\mathbb{N}_{J}$ and $a\in\\{0,1\\}$.
Since we are arguing convergence in probability, we can treat each of these
components separately. To that end let $\varepsilon>0$ be given, and fix
$s\in\mathbb{N}_{\mathcal{S}}$ $a\in\\{0,1\\}$ and
$j\in\mathbb{N}_{\mathcal{J}}$. Our argument will take the following form.
Suppose $R_{n}^{\ast}(a,s,j)$ is an auxiliary random variable with the same
distribution as $R_{n}(a,s,j)$. Furthermore, suppose that we have an arbitrary