# Regret Analysis in Threshold Policy Design††thanks: I am grateful to Ivan
Canay, Joel Horowitz and Charles Manski for their guidance in this project. I
am also thankful to Anastasiia Evdokimova, Amilcar Velez and all the
participants of the Econometric Reading Group at Northwestern University for
their comments and suggestions.

Federico Crippa
Department of Economics
Northwestern University
<EMAIL_ADDRESS>
(April 17, 2024)

Threshold policies are targeting mechanisms that assign treatments based on
whether an observable characteristic exceeds a certain threshold. They are
widespread across multiple domains, such as welfare programs, taxation, and
clinical medicine. This paper addresses the problem of designing threshold
policies using experimental data, when the goal is to maximize the population
welfare. First, I characterize the regret—a measure of policy optimality—of
the Empirical Welfare Maximizer (EWM) policy, popular in the literature. Next,
I introduce the Smoothed Welfare Maximizer (SWM) policy, which improves the
EWM’s regret convergence rate under an additional smoothness condition. The
two policies are compared studying how differently their regrets depend on the
population distribution, and investigating their finite sample performances
through Monte Carlo simulations. In many contexts, the welfare guaranteed by
the novel SWM policy is larger than with the EWM. An empirical illustration
demonstrates how the treatment recommendation of the two policies may in
practice notably differ.

Keywords: Threshold policies, heterogeneous treatment effects, statistical
decision theory, randomized experiments.

JEL classification codes: C14, C44

##  1 Introduction

Treatments are rarely universally assigned. When their effects are
heterogeneous across individuals, policymakers aim to target those who would
benefit the most from specific interventions. Scholarships, for example, are
awarded to students with high academic performance or financial need; tax
credits are provided to companies engaged in research and development
activities; medical treatments are prescribed to sick patients. Despite the
potential complexity and multidimensionality of heterogeneous treatment
effects, treatment eligibility criteria are often kept quite simple. This
paper studies one of the most common of these simple assignment mechanisms:
threshold policies, where the decision to assign the treatment is based on
whether a scalar observable characteristic — referred to as the index —
exceeds a specified threshold.

Threshold policies are ubiquitous, ranging across multiple domains. In welfare
policies, they regulate the qualification for public health insurance programs
through age (Card et al., 2008; Shigeoka, 2014) and anti-poverty programs
through income (Crost et al., 2014). In taxation, they determine marginal
rates through income brackets (Taylor, 2003). In clinical medicine, the
referral for liver transplantation depends on whether a composite of
laboratory values obtained from blood tests is beyond a certain threshold
(Kamath and Kim, 2007). Even criminal offenses are defined through threshold
policies: sanctions for Driving Under the Influence are based on whether the
Blood Alcohol Content exceeds specific values.

The regression discontinuity design has been developed to study the treatment
effect at the point of discontinuity of threshold policies: its popularity in
econometrics and applied economics is a further indication of how widespread
threshold policies are. Regression discontinuity design focuses on an ex-post
evaluation. In this paper, my perspective is different: I consider the ex-ante
problem faced by a policymaker wanting to implement a threshold policy and
interested in maximizing the average social welfare, targeting individuals who
would benefit from the treatment. Experimental data are available: how should
they be used to implement the threshold policy in the population?

Answering this question requires defining a criterion by which policies are
evaluated. Since the performance of a policy depends on the unknown data
distribution, the policy maker searches for a policy that behaves uniformly
well across a specified family of data distribution (the state space). The
regret of a policy is the (possibly) random difference between the maximum
achievable welfare and the welfare it generates in the population. Policies
can be evaluated considering their maximum expected regret (Manski, 2004;
Hirano and Porter, 2009; Kitagawa and Tetenov, 2018; Athey and Wager, 2021;
Mbakop and Tabord-Meehan, 2021; Manski, 2023), or other worst-case statistics
of the regret distribution (Manski and Tetenov, 2023; Kitagawa et al., 2022).
Once the criterion has been established, optimal policy learning aims to
pinpoint the policy that minimizes it. Rather than directly tackling the
functional minimization problem, following the literature, I consider
candidate threshold policy functions and characterize some properties of their
regret.

The first contribution of this paper is to show how to derive the asymptotic
distribution of the regret for a given threshold policy. The underlying
intuition is simple: threshold policies use sample data to choose the
threshold, which is hence a random variable with a certain asymptotic
behavior. A Taylor expansion establishes a map between the regret of a policy
and its threshold, allowing one to characterize the asymptotic distribution of
the regret through the asymptotic behavior of the threshold. This shifts the
problem to characterizing the asymptotic of the threshold estimator,
simplifying the analysis as threshold estimators can be studied with common
econometric tools.

I start considering the Empirical Welfare Maximizer (EWM) policy studied by
kitagawa2018should. They derive uniform bounds for the expected regret of the
policy for various policy classes, where the policy class impacts the findings
only in terms of its VC dimensionality. My approach is more specific,
considering only threshold policies, but also more informative: leveraging the
knowledge of the policy class, I characterize the entire asymptotic
distribution of the regret. As mentioned above, this requires the derivation
of the asymptotic distribution of the threshold for the EWM policy, which is
non-standard: it exhibits the “cube root asymptotics” behavior studied in
kim1990cube. The convergence rate is $n^{1/3}$, and the asymptotic
distribution is of chernoff1964estimation type. The non-standard behavior and
the unusual convergence rate are due to the discontinuity in the objective
function and are reflected in the asymptotic distribution of the regret, and
in its $n^{2/3}$ convergence rate.

My second contribution is hence to propose a novel threshold policy, the
Smoothed Welfare Maximizer (SWM) policy. This policy replaces the indicator
function in the EWM policy’s objective function with a smooth kernel. Under
certain regularity assumptions, the threshold estimator for the SWM policy is
asymptotically normal and its regret achieves a $n^{4/5}$ convergence rate.
This implies that the regret’s convergence rate with the SWM is faster than
with the EWM policy.

Building on these asymptotic results, I extend the comparison of the regrets
with the EWM and the SWM policies beyond their convergence rates. My findings
allow to compare the asymptotic distributions and investigate how differently
they depend on the data distribution; theoretical results are helpful to
inform and guide the Monte Carlo simulations, which confirm that the
asymptotic results approximate the actual finite sample behaviors. Notably,
the simulations confirm that the SWM policy may guarantee lower expected
regret in finite samples.

To demonstrate the practical differences between the two policies, I present
an empirical illustration considering a job-training treatment. In that
context, the SWM threshold policy would recommend treating 79% of unemployed
workers, as opposed to 83% with the EWM policy. This difference of 4
percentage points is economically non-negligible.

###  1.1 Related Literature

This paper relates to the statistical decision theory literature studying the
problem of policy assignment with covariates (manski2004statistical;
stoye2012minimax; kitagawa2018should; athey2021policy; mbakop2021model;
sun2021treatment; sun2021empirical; viviano2023fair). My setting is mainly
related to kitagawa2018should and athey2021policy, with some notable
differences. kitagawa2018should study the EWM policy for policy classes with
finite VC dimension. They derive finite sample bounds for regret without
relying on smoothness assumptions. athey2021policy consider a double robust
version of the EWM and allow for observational data. Under smoothness
assumptions analogous to mine, they derive asymptotic bounds for the expected
regret for policy classes with finite VC dimensions. Conversely, results in
this paper apply exclusively to threshold policies, relying on a combination
of the assumptions in kitagawa2018should and athey2021policy. The narrower
focus allows for more comprehensive results: the entire asymptotic
distribution of the regret is derived rather than providing some bounds for
the expected regret.

Another critical distinction lies in the different nature of the uniform
convergence rates: my state space is a subset of theirs, which is why the
rates I derive for the EWM and the SWM policies are faster than the $\sqrt{n}$
rate reported as the optimal by kitagawa2018should and athey2021policy. Their
$\sqrt{n}$ rate is, in fact, uniformly valid for a family of data
distributions that, at least for threshold policies, include extreme cases
(e.g. discontinuous conditional ATE), which determine the slower rate of
convergence. Their uniform results may be viewed as a benchmark: when more
structure is added to the problem and some distribution in the family
excluded, they can be improved.

Optimal policy learning finds its empirical counterpart in the literature
dealing with targeting, especially common in development economics. Recent
studies rely on experimental evidence to decide who to treat in a data-driven
way (hussam2022targeting; aiken2022machine), even if haushofer2022targeting
pointed out the need for a more formalized approach to the targeting decision
problem. The availability for the policy maker of appropriate tools to use the
data in the decision process is probably necessary to guarantee a broader
adoption of data-driven targeting strategies. Focusing on threshold policies,
this paper explicitly formulates the decision problem, introduces
implementable policies (the EWM and the SWM policy), and compares their
asymptotic properties.

Turning to the threshold estimators, I already mentioned that the EWM policy
exhibits the cube root of $n$ asymptotics studied by kim1990cube, distinctive
of several estimators in different contexts. Noteworthy examples are the
maximum score estimator in choice models (manski1975maximum), the split point
estimator in decision trees (banerjee2007confidence), and the risk minimizer
in classification problems (mohammadi2005asymptotics), among others. Specific
to my analysis is the emergence of the cube root asymptotic within a causal
inference problem relying on the potential outcomes model, which is then
mirrored in the regret’s asymptotic distribution.

Addressing the cube root problem by smoothing the indicator in the objective
function aligns closely with the strategy proposed by horowitz1992smoothed for
studying the asymptotic behavior of the maximum score estimator. Objective
functions are nonetheless different, and in my context, I derive the
asymptotic distribution for both the unsmoothed (EWM) and the smoothed (SWM)
policies. This is convenient, as it allows me to compare not only the
convergence rates but also the entire asymptotic distributions of the
estimators and their regrets and study the asymptotic approximations in Monte
Carlo simulations.

The rest of the paper is structured as follows. Section 2 introduces the
problem and outlines my analytical approach. Section 3 derives formal results
for the asymptotic of the EWM and SWM policies and their regrets. In Section
4, I investigate finite sample performance of the EWM and SWM policies through
Monte Carlo simulations, while in Section 5 I consider the analysis of
experimental data from the National Job Training Partnership Act (JTPA) Study
to compare the practical implications of the policies. Section 6 concludes.

##  2 Overview of the Problem

I consider the problem of a policymaker who wants to implement a binary
treatment in a population of interest. Individuals are characterized by a
vector of observable characteristics $\textbf{X}\in\mathbb{R}^{d}$, on which
the policy maker bases the treatment assignment choice. A policy is hence a
map $\pi(\textbf{x}):\mathbb{R}^{d}\rightarrow\\{0,1\\}$, from observable
characteristics to the binary treatment status. The policy maker is
utilitarian: its goal is to maximize the average welfare in the population.
Indicating by $Y_{1}$ and $Y_{0}$ the potential outcomes with and without the
treatment, population average welfare generated by a policy $\pi$ can be
written as

$\displaystyle W(\pi)=\mathbb{E}[Y_{1}\pi(X)+Y_{0}(1-\pi(X))].$ (1)

When treatment effects are heterogeneous, the same treatment can have opposite
average effects across individuals with different X. For this reason, the
policy assignment may vary with X: the policymaker wants to target only those
who benefit from being treated, to maximize the average welfare.

The policy learning literature has considered several classes $\Pi$ of policy
functions, such for example linear eligibility indexes, decision trees, or
monotone rules, discussed in kitagawa2018should; athey2021policy;
mbakop2021model. This paper focuses on threshold policies, a specific class of
policy functions that can be represented as

$\displaystyle\pi(\textbf{X})=\pi(X,t)=\mathbf{1}\\{X>t\\}.$ (2)

Treatment is assigned whenever the scalar observable index $X\in\mathbb{R}$
exceeds threshold $t$, a parameter that must be chosen. These threshold
policies are widespread: they regulate, beyond others, organ transplants
kamath2007model, taxation (taylor2003corporation), and access to social
welfare programs (card2008impact; crost2014aid). This paper aims not to
justify or advocate for the use of threshold policies, and the restriction to
this policy class is taken as exogenous.

I will focus on the case when the index $X$ is chosen before the experiment.
Population welfare depends only on threshold $t$, and can be written as

$\displaystyle
W(\pi)=W(t)=\mathbb{E}[Y_{1}\mathbf{1}\\{X>t\\}+Y_{0}\mathbf{1}\\{X\leq
t\\}].$ (3)

Choosing the policy is equivalent to choosing the threshold. If the joint
distribution of $Y_{1}$, $Y_{0}$, and $X$ were known, the policy maker would
implement the policy with threshold $t^{*}$ defined as:

$\displaystyle t^{*}\in\mathop{\rm
arg~{}max}\limits_{t}\mathbb{E}[Y_{1}\mathbf{1}\\{X>t\\}+Y_{0}\mathbf{1}\\{X\leq
t\\}]$ (4)

which would guarantee the maximum achievable welfare $W(t^{*})$.

The problem described in equation (4) is unfeasible since the joint
distribution of $Y_{1}$, $Y_{0}$, and $X$ is unknown. The policy maker
observes an experimental sample
$Z=\\{Z_{i}\\}_{i=1}^{n}=\\{Y_{i},D_{i},X_{i}\\}$, where $Y$ is the outcome of
interest, $D$ the randomly assigned treatment status, and $X$ the policy
index. Experimental data, which allows to identify the conditional average
treatment effect, are used to learn the threshold policy
$\hat{t}_{n}=\hat{t}_{n}(Z)$, function of the observed sample.

Statistical decision theory deals with the problem of choosing the map
$\hat{t}_{n}$. First, it is necessary to specify the decision problem the
policymaker faces. For any threshold policy $\hat{t}_{n}$, define the regret
$\mathcal{R}(\hat{t}_{n})$:

$\displaystyle\mathcal{R}(\hat{t}_{n})=W(t^{*})-W(\hat{t}_{n}),$ (5)

a measure of welfare loss indicating the suboptimality of policy
$\hat{t}_{n}$. The regret depends on the unknown data distribution: the
policymaker specifies a state space, and searches for a policy that does well
uniformly for all the data distributions in the state space. Following
manski2004statistical, statistical decision theory has mainly focused on the
problem of minimizing the maximum expected regret, looking for a policy
$\hat{t}_{n}$ that does uniformly well on average across repeated samples.

Directly solving the constrained minimization problem of the functional
$\sup\mathbb{E}[\mathcal{R}(\hat{t}_{n})]$ is impractical: the literature
instead focuses on considering a specific policy map and studying its
properties, for example showing its rate optimality, through finite sample
valid (kitagawa2018should) or asymptotic (athey2021policy) arguments.
Following this approach, I characterize and compare some properties for the
regret of two different threshold policies, the Empirical Welfare Maximizer
(EWM) policy, commonly studied in the literature, and the novel Smoothed
Welfare Maximizer (SWM) policy.

kitagawa2018should derive bounds for the expected regret of the EWM policy for
a wide range of policy function classes. In their results, the policy class
$\Pi$ affects the bounds only through its VC dimension, and the knowledge of
$\Pi$ is not further exploited. Conversely, I leverage the additional
structure from the knowledge of the policy class and characterize the entire
asymptotic distribution of the regret for the EWM and the SWM threshold
policies, comparing how their regrets depend on the data distribution. My
results could hence be of interest also when decision problems not involving
the expected regret are considered, as in manski2023statistical and
kitagawa2022treatment: I characterize the asymptotic behavior of regret
quantiles, and the asymptotic distributions can be used to simulate
expectations of their non-linear functions.

To derive my results, I take advantage of the link between a threshold policy
function $\hat{t}_{n}$ and its regret $\mathcal{R}(\hat{t}_{n})$. Let
$\\{r_{n}\\}$ be a sequence such that $r_{n}\rightarrow\infty$ for
$n\rightarrow\infty$, and suppose that $r_{n}(\hat{t}_{n}-t^{*})$ converges to
a non degenerate limiting distribution, i.e
$(\hat{t}_{n}-t^{*})=O_{p}(r_{n}^{-1})$.

Assume function $W(t)$ to be twice continuously differentiable, and consider
its second-order Taylor expansion around $t^{*}$:

$\displaystyle
W(\hat{t}_{n})=W(t^{*})+\underbrace{W^{\prime}(t^{*})}_{=0}\left(\hat{t}_{n}-t^{*}\right)+\frac{1}{2}W^{\prime\prime}(\tilde{t})\left(\hat{t}_{n}-t^{*}\right)^{2}$

where $|\tilde{t}-t^{*}|\leq|\hat{t}_{n}-t^{*}|$, and $W^{\prime}(t^{*})=0$ by
optimality of $t^{*}$. The previous equation can be written as

$\displaystyle
r_{n}^{2}\mathcal{R}(\hat{t}_{n})=\frac{1}{2}W^{\prime\prime}(\tilde{t})\left[r_{n}\left(\hat{t}_{n}-t^{*}\right)\right]^{2},$
(6)

establishing a relationship between the convergence rates of $\hat{t}_{n}$ and
$\mathcal{R}(\hat{t}_{n})$, and between their asymptotic distributions.
Equation (6) therefore shows how the rate of convergence and the asymptotic
distribution of regret $\mathcal{R}(\hat{t}_{n})$ can be studied through the
rate of convergence and the asymptotic distribution of policy $\hat{t}_{n}$.
In the next section, I consider the EWM policy $\hat{t}^{e}_{n}$ and the SWM
policy $\hat{t}^{s}_{n}$: through their asymptotic behaviors, I characterize
the asymptotic distributions of their regrets $\mathcal{R}(\hat{t}^{e}_{n})$
and $\mathcal{R}(\hat{t}^{s}_{n})$.

##  3 Formal Results

Let $Y_{0}$ and $Y_{1}$ be scalar potential outcomes, $D$ the binary treatment
assignment in the experiment, $\textbf{X}\in\mathbb{R}^{d}$ a vector of $d$
observable characteristics, and $X$ the observable index.
$\\{Y_{0},Y_{1},D,\textbf{X}\\}$ are random variables distributed according to
the distribution $P$. They satisfy the following assumptions, which guarantee
the identification of the optimal threshold:

######  Assumption 1.

(Identification)

  1. 1.1

(No interference) Observed outcome $Y$ is related with potential outcomes by
the expression $Y=DY_{1}+(1-D)Y_{0}$.

  2. 1.2

(Unconfoundedness) Distribution $P$ satisfies
$D\perp\\!\\!\\!\perp(Y_{0},Y_{1})|\textbf{X}$.

  3. 1.3

(Overlap) Propensity score $p(\textbf{x})=\mathbb{E}[D|\textbf{X}=\textbf{x}]$
is assumed to be known and such that $p(\textbf{x})\in(\eta,1-\eta)$, for some
$\eta\in(0,0.5)$.

  4. 1.4

(Joint distribution) Potential outcomes $(Y_{0},Y_{1})$ and index $X$ are
continuous random variables with joint probability density function
$\varphi(y_{0},y_{1},x)$, and marginal densities $\varphi_{0}$, $\varphi_{1}$,
and $f_{x}$ respectively. Expectations $\mathbb{E}[Y_{0}|x]$ and
$\mathbb{E}[Y_{1}|x]$, for $x$ in the support of $X$, exist.

Assumptions 1.1, 1.2, and 1.3 are standard assumptions in many causal models.
Assumption 1.1 requires the outcome of each unit to depend only on their
treatment status, excluding spillover effects. Assumption 1.2 requires random
assignment of the treatment, conditionally on X. Assumption 1.3 requires that,
for any value of X, there is a positive probability of observing both treated
and untreated units. Probabilities of being assigned to the treatment may vary
with X, allowing for stratified experiments.

Assumption 1.4 specifies the focus on continuous outcome and index. While it
would be possible to accommodate discrete $Y_{0}$ and $Y_{1}$, maintaining the
continuity of $X$ remains essential. The arguments developed in this paper, in
fact, are not valid for a discrete index: my focus is on studying optimal
threshold policies in contexts where the probability of observing any value on
the support of the index $X$ is zero, and the threshold must be chosen from a
continuum of possibilities.

Under Assumption 1, optimal policy $t^{*}$ defined in (4) can be written as

$\displaystyle t^{*}\in$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\mathbb{E}_{P}[Y_{1}\mathbf{1}\\{X>t\\}+Y_{0}\mathbf{1}\\{X\leq
t\\}]$ (7) $\displaystyle=$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\mathbb{E}_{P}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\mathbf{1}\\{X>t\\}\right]$
(8)

and is hence identified. This standard result specifies under which conditions
an experiment allows to identify $t^{*}$.

###  3.1 Empirical Welfare Maximizer Policy

Policymaker observes an i.i.d. random sample $Z=\\{Y_{i},D_{i},X_{i}\\}$ of
size $n$ from $P$, and considers the Empirical Welfare Maximizer policy
$\hat{t}^{e}_{n}$, the sample analog of $t^{*}$ in equation
(4)111Computationally, the problem requires to evaluate the function inside
the argmax $n+1$ times; the solution is the convex set of points in
$\mathbb{R}$ that give the maximum of these values.:

$\displaystyle\hat{t}^{e}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}\mathbf{1}\\{X_{i}>t\\}+\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\mathbf{1}\\{X_{i}\leq
t\\}\right).$ (9)

Policy $\hat{t}^{e}_{n}$ can be seen as an extremum estimator, maximizer of a
function not continuous in $t$.

####  3.1.1 Consistency of $\hat{t}^{e}_{n}$

First, I will prove that $\hat{t}^{e}_{n}$ consistently estimates the optimal
threshold $t^{*}$, implying that
$\mathcal{R}(\hat{t}^{e}_{n})\rightarrow^{p}0$. To prove this result, I need
the following assumptions on the data distribution.

######  Assumption 2.

(Consistency)

  1. 2.1

(Maximizer $t^{*}$) Maximizer $t^{*}\in\mathcal{T}$ of
$\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ exists and is unique. It is an
interior point of the compact parameter space
$\mathcal{T}\subseteq\mathbb{R}$.

  2. 2.2

(Square integrability) Conditional expectations $\mathbb{E}[Y_{0}^{2}|X]$ and
$\mathbb{E}[Y_{1}^{2}|X]$ exist.

  3. 2.3

(Smoothness) In a neighbourhood of $t^{*}$, density $f_{x}(x)$ is positive,
and function $\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ is at least
$s$-times continuously differentiable in $t$.

By requiring the existence of the optimal threshold in the interior of the
parameter space, Assumption 2.1 is assuming heterogeneity in the sign of the
conditional average treatment effect $\mathbb{E}[Y_{1}-Y_{0}|X]$. It is
because of this heterogeneity that the policymaker implements the threshold
policy, targeting groups that would benefit from being treated. The assumption
neither excludes the multiplicity of local maxima, as long as the global one
is unique, nor excludes unbounded support for $X$, but requires the parameter
space to be compact. A sufficient condition for Assumption 2.1, easy to
interpret and plausible in many applications, is that the conditional average
treatment effect has negative and positive values, and crosses zero exactly
once.

Assumption 2.2 requires that the conditional potential outcomes have finite
second moments and is satisfied when $Y$ is assumed to be bounded (as in
kitagawa2018should).

Assumption 2.3 will be used with increasing values of $s$ to prove different
results. To prove consistency, it needs to hold for $s=0$, requiring the
continuity of the objective function $W(t)$ in a neighborhood of $t^{*}$. The
derivative of $\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ with respect to
$t$ is equal to $f_{x}(t)\tau(t)$, where $\tau(x)=\mathbb{E}[Y_{1}-Y_{0}|X=x]$
is the conditional average treatment effect. Assumption 2.3 with $s\geq 1$
hence requires smoothness of $f_{x}(x)$ and $\tau(x)$, in a neighborhood of
$t^{*}$.

The following theorem proves the consistency of $\hat{t}^{e}_{n}$ for $t^{*}$.

######  Theorem 1.

Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1 and 2 with
$s=0$,

$\displaystyle\hat{t}^{e}_{n}\rightarrow^{a.s.}t^{*}$

i.e. $\hat{t}^{e}_{n}$ is a consistent estimator for $t^{*}$.

Note that, since $\hat{t}^{e}_{n}$ is not continuous in $t$, the proof of
Theorem 1 does not resort to the standard arguments for consistency of
extremum estimators. Conversely, it relies on the approach used to prove
consistency for estimators exhibiting the “cube root asymptotics” behavior,
discussed in the next section.

####  3.1.2 Asymptotic Distribution for $\hat{t}^{e}_{n}$

The fact that $\hat{t}^{e}_{n}$ is not continuous in $t$ directly affects the
convergence rate and the asymptotic distribution. The EWM policy
$\hat{t}^{e}_{n}$ exhibits the “cube root asymptotics” behavior studied in
kim1990cube, the same as, beyond others, the maximum score estimator
(manski1975maximum), and the split point estimator in decision trees
(banerjee2007confidence).

The limiting distribution is not Gaussian, and its derivation requires an
additional regularity condition on the tails of the probability density
distribution of $Y_{1}$ and $Y_{0}$:

######  Assumption 3.

(Tail condition) Let $\varphi_{1}$ and $\varphi_{0}$ be the probability
density functions of $Y_{1}$ and $Y_{0}$. Assume that, as
$|y|\rightarrow\infty$, $\varphi_{1}(y)=o(|y|^{-(4+\delta)})$ and
$\varphi_{0}(y)=o(|y|^{-(4+\delta)})$, for $\delta>0$.

The following theorem gives the asymptotic distribution of $\hat{t}^{e}_{n}$.

######  Theorem 2.

Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=2$ and 3, as $n\rightarrow\infty$,

$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right)$ (10)

where $B(r)$ is the two-sided standard Brownian motion process, and $K$ and
$H$ are

$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
$\displaystyle H=$ $\displaystyle
f_{x}(t^{*})\left(\frac{\partial\mathbb{E}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$

The limiting distribution of $n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)$ is a
of Chernoff type (chernoff1964estimation). The Chernoff’s distribution is the
probability distribution of the random variable $\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}$, where $B(r)$ is the two-sided standard
Brownian motion process. The process $B(r)-r^{2}$ can be simulated, and the
distribution of $\mathop{\rm arg~{}max}\limits_{r}B(r)-r^{2}$ numerically
studied. groeneboom2001computing report values for selected quantiles.

It’s worth noticing how the variance of $\hat{t}^{e}_{n}$ depends on the data
distribution. $K$ and $H$ are functions of the density of $X$, the variance of
the Conditional Average Treatment Effect, and the derivative of the CATE at
$t^{*}$. The optimal threshold is estimated with more precision when more data
around the optimal threshold are available (larger density), when the
treatment effect changes more rapidly (larger derivative of CATE), and when
the outcomes have less variability (smaller variance of CATE). Exponents on
$K$ and $H$, determined by the cube root behavior, will be crucial for
comparing $\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$.

Results in Theorem 2 can be used to derive asymptotic valid confidence
intervals for $\hat{t}^{e}_{n}$, as discussed in appendix A. More
interestingly, they can be combined with equation 6 to characterize the
asymptotic distribution of the regret $\mathcal{R}(\hat{t}^{e}_{n})$, as
derived in the following corollary.

######  Corollary 2.1.

The asymptotic distribution of regret $\mathcal{R}(\hat{t}^{e}_{n})$ is:

$\displaystyle
n^{\frac{2}{3}}\mathcal{R}(\hat{t}^{e}_{n})\rightarrow^{d}\left(\frac{2K^{2}}{H}\right)^{\frac{1}{3}}\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}.$

The expected value of the asymptotic distribution is
$K^{\frac{2}{3}}H^{-\frac{1}{3}}C^{e}$, where

$C^{e}=\sqrt[3]{2}\mathbb{E}\left[\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}\right]$

is a constant not dependent on $P$.

For the regret of the EWM policy, Corollary 2.1 establishes a
$n^{\frac{2}{3}}$ rate, faster than the $\sqrt{n}$ rate found to be the
optimal for the EWM expected regret (kitagawa2018should). It is essential to
highlight how the two results are different: the rate derived by
kitagawa2018should for the expected regret is valid uniformly over a family of
distributions that may violate the assumptions in Theorem 4, for example
including distributions where the CATE is discontinuous at the threshold. On
the other hand, Corollary 2.1 is derived for distributions satisfying
Assumptions 1, 2 and 3. They imply that $K$ is bounded and $H$ is bounded away
from zero: the state space can be further characterized setting an upper bound
$\overline{K}<\infty$ for $K$ and a lower bound $\underline{H}>0$ for $H$,
implying a maximum for the expectation of the asymptotic regret equal to
$\overline{K}^{\frac{2}{3}}\underline{H}^{-\frac{1}{3}}C^{e}$.

###  3.2 Smoothed Welfare Maximizer Policy

Corollary 2.1 shows how the cube root of $n$ convergence rate of the EWM
policy directly impacts the convergence rate of its regret. In this section, I
propose an alternative threshold policy, the Smoothed Welfare Maximizer
policy, that achieves a faster rate of convergence and hence guarantees a
faster rate of convergence for its regret. My approach exploits some
additional smoothness assumptions on the distribution $P$: Corollary 2.1 holds
when $f_{x}(x)$ and $\tau(x)$ are assumed to be at least once differentiable;
if they are at least twice differentiable, the SWM policy guarantees a
$n^{\frac{4}{5}}$ convergence rate for the regret. Note that asking the
density of the index and the conditional average treatment effect to be twice
differentiable seems plausible for many applications. In the context of policy
learning, it is, for example, assumed by athey2021policy to derive their
results222This assumption is implied by the high-level assumptions made in the
paper, as the authors discuss in footnote 15..

My approach involves smoothing the objective function in (9), in the same
spirit as the smoothed maximum score estimator proposed by
horowitz1992smoothed to deal with inference for the maximum score estimator
(manski1975maximum). The Smoothed Welfare Maximizer (SWM) policy
$\hat{t}^{s}_{n}$ is defined as:

$\displaystyle\hat{t}^{s}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right]$
(11)

where $\sigma_{n}$ is a sequence of positive real numbers such that
$\lim_{n\rightarrow\infty}\sigma_{n}=0$, and the function $k(\cdot)$
satisfies:

######  Assumption 4.

(Kernel function) Kernel function $k(\cdot):\mathbb{R}\rightarrow\mathbb{R}$
is continuous, bounded, and with limits $\lim_{x\rightarrow-\infty}k(x)=0$ and
$\lim_{x\rightarrow\infty}k(x)=1$.

In practice, the indicator function found in $\hat{t}^{e}_{n}$ is here
substituted by a smooth function $k(\cdot)$ with the same limiting behavior,
which guarantees the differentiability of the expression in the argmax. The
bandwidth $\sigma_{n}$, decreasing with the sample size, ensures that when
$n\to\infty$ the policy converges to the optimal one, as proved in the next
section.

####  3.2.1 Consistency of $\hat{t}^{s}_{n}$

I start showing consistency of $\hat{t}^{s}_{n}$ for $t^{*}$, which implies
$\mathcal{R}(\hat{t}^{s}_{n})\rightarrow^{p}0$.

######  Theorem 3.

Consider the SWM policy $\hat{t}^{s}_{n}$ defined in equation (11) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=0$ and 4, as $n\rightarrow\infty$,

$\displaystyle\hat{t}^{s}_{n}\rightarrow^{a.s.}t^{*}$

i.e. $\hat{t}^{s}_{n}$ is a consistent estimator for $t^{*}$.

Theorems 3 and 1 are analogous: they rely on the same assumptions on the data
(Assumptions 1, 2) to prove the consistency of $\hat{t}^{e}_{n}$ and
$\hat{t}^{s}_{n}$. Where the two policies differ is in the asymptotic
distributions: smoothness in the objective function for $\hat{t}^{s}_{n}$
guarantees asymptotic normality, but also introduces a bias, since the
bandwidth $\sigma_{n}$ equals zero only in the limit, which pops up in the
limiting distribution.

####  3.2.2 Asymptotic Distribution for $\hat{t}^{s}_{n}$

Deriving this asymptotic behavior of $\hat{t}^{s}_{n}$ requires an additional
assumption on the rate of bandwidth $\sigma_{n}$ and the kernel function $k$.
Since both are chosen by the policy maker, the assumption is not a restriction
on unobservables but a condition on properly picking $\sigma_{n}$ and $k$.

######  Assumption 5.

(Bandwidth and kernel)

  1. 5.1

(Rate of $\sigma_{n}$) $\frac{\log n}{n\sigma_{n}^{4}}\rightarrow 0$ as
$n\rightarrow\infty$.

  2. 5.2

(Kernel function) Kernel function $k(\cdot):\mathbb{R}\rightarrow\mathbb{R}$
satisfies Assumption 4 and the following:

     * •

$k(\cdot)$ is continuous, bounded, and with limits
$\lim_{x\rightarrow-\infty}k(x)=0$ and $\lim_{x\rightarrow\infty}k(x)=1$.

     * •

$k(\cdot)$ is twice differentiable, with uniformly bounded derivatives
$k^{\prime}$ and $k^{\prime\prime}$.

     * •

$\int k^{\prime}(x)^{4}dx$, $\int k^{\prime\prime}(x)^{2}dx$, and
$\int|x^{2}k^{\prime\prime}(x)|dx$ are finite.

     * •

For some integer $h\geq 2$ and each integer $i\in[1,h]$,
$\int|x^{i}k^{\prime}(x)|dx=0$ for
$i<h$ and $\int|x^{h}k^{\prime}(x)|dx=d\neq 0$, with $d$ finite.

     * •

For any integer $i\in[0,h]$, any $\eta>0$, and any sequence
$\sigma_{n}\rightarrow 0$,
$\lim_{n\rightarrow\infty}\sigma_{n}^{i-h}\int_{|\sigma_{n}x|>\eta}|x^{i}k^{\prime}(x)|dx=0$,
and
$\lim_{n\rightarrow\infty}\sigma_{n}^{-1}\int_{|\sigma_{n}x|>\eta}|k^{\prime\prime}(x)|dx=0$.

     * •

$\int xk^{\prime\prime}(x)dx=1$,
$\lim_{n\rightarrow\infty}\int_{|\sigma_{n}x|>\eta}|xk^{\prime\prime}(x)|dx=0$.

An example of a function $k$ satisfying Assumption 5.2 with $h=2$ is the
cumulative distribution function of the standard normal distribution.

I can now derive the asymptotic distribution of $\hat{t}^{s}_{n}$.

######  Theorem 4.

Consider the SWM policy $\hat{t}^{s}_{n}$ defined in equation (11) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=h+1$ for some $h\geq 2$, and 5, as $n\rightarrow\infty$:

  1. 1.

if $n\sigma_{n}^{2h+1}\rightarrow\infty$,

$\sigma_{n}^{-h}(\hat{t}^{s}_{n}-t^{*})\rightarrow^{p}H^{-1}A;$

  2. 2.

if $n\sigma_{n}^{2h+1}\rightarrow\lambda<\infty$,

$(n\sigma_{n})^{1/2}(\hat{t}^{s}_{n}-t^{*})\rightarrow^{d}\mathcal{N}(\lambda^{1/2}H^{-1}A,H^{-2}\alpha_{2}K);$

where $A$, $\alpha_{1}$, and $\alpha_{2}$ are:

$\displaystyle A=$
$\displaystyle-\frac{1}{h!}\alpha_{1}\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{x}(y,t^{*})dy$
(12) $\displaystyle\alpha_{1}=$
$\displaystyle\int_{\zeta}\zeta^{h}k^{\prime}\left(\zeta\right)d\zeta$ (13)
$\displaystyle\alpha_{2}=$
$\displaystyle\int_{\zeta}k^{\prime}\left(\zeta\right)^{2}d\zeta.$ (14)

The asymptotic distribution of $(n\sigma_{n})^{1/2}(\hat{t}^{s}_{n}-t^{*})$ is
normal, centered at the asymptotic bias $\lambda^{1/2}H^{-1}A$ introduced by
the smoothing function, which exploits local information giving non-zero
weights to treated units in the untreated region and vice versa. The bias and
the variance of the distribution depend on the population distribution through
$K$ and $H$, as for the EWM policy, and also through $A$, a new term that
determines the bias. In the definition of $A$, $\varphi^{h}_{x}$ is the $h$
derivative with respect to $x$ of $\varphi(y_{0},y_{1},x)$, the joint density
distribution of $Y_{0}$, $Y_{1}$, and $X$: the integral in the expression for
$A$ is the expectation of the $h$-derivative of $f_{x}(X)\tau(X)$ computed in
$X=t^{*}$, whose existence is guaranteed by Assumption 2.3 with $s=h+1$.
$\alpha_{1}$ and $\alpha_{2}$ depends only on kernel function $k$, and are
hence known.

The difference between Theorems 4 and 2 is the degree of smoothness required
by Assumption 2.3. In Theorem 2, the objective function $W(t)$ is required to
be twice differentiable, while Theorem 4 requires $h+1$ continuous
derivatives, where $h$ directly impacts rate of convergence of the policy: to
achieve a rate faster than the EWM policy, $\hat{t}^{s}_{n}$ requires $W(t)$
to be three times differentiable.

As for the EWM policy, results in Theorem 4 can be used to derive asymptotic
valid confidence intervals for $\hat{t}^{s}_{n}$ (see appendix A), and,
combined with equation 6, to characterize the asymptotic distribution of the
regret $\mathcal{R}(\hat{t}^{s}_{n})$, as derived in the next corollary.

######  Corollary 4.1.

Asymptotic distribution of regret $\mathcal{R}(\hat{t}^{s}_{n})$ is:

$\displaystyle
n\sigma_{n}\mathcal{R}(\hat{t}^{s}_{n})\rightarrow^{d}\frac{1}{2}\frac{\alpha_{2}K}{H}\chi^{2}\left(1,\frac{\lambda
A^{2}}{\alpha_{2}K}\right)$

where $\chi^{2}\left(1,\frac{\lambda A^{2}}{\alpha_{2}K}\right)$ is a non-
centered chi-squared distribution with 1 degree of freedom and non-central
parameter $\frac{\lambda A^{2}}{\alpha_{2}K}$. The expected value of the
asymptotic distribution is:

$\displaystyle\frac{1}{2}\frac{\alpha_{2}K}{H}\left(1+\frac{\lambda
A^{2}}{\alpha_{2}K}\right)=\frac{\alpha_{2}}{2}\frac{K}{H}+\frac{1}{2}\frac{\lambda
A^{2}}{H}.$ (15)

Let $\sigma_{n}=(\lambda/n)^{1/(2h+1)}$ with $\lambda\in(0,\infty)$. The
expectation of the asymptotic regret is minimized by setting
$\lambda=\lambda^{*}=\frac{\alpha_{2}K}{2hA^{2}}$: in this case, the
expectation of the asymptotic distribution scaled by $n^{\frac{2h}{2h+1}}$ is
$A^{\frac{2}{2h+1}}K^{\frac{2h}{2h+1}}H^{-1}C^{s}$, where
$C^{s}=\frac{2h+1}{2}\left(\frac{\alpha_{2}}{2h}\right)^{\frac{2h}{2h+1}}$ is
a constant not dependent on $P$.

With the optimal bandwidth $\sigma_{n}=O_{p}(n^{-\frac{1}{2h+1}})$, the regret
converges at $n^{\frac{2h}{2h+1}}$ rate. For $h\geq 2$, this implies that the
regret converges faster with the SWM than with the EWM policy: the extra
smoothness assumption has been exploited to achieve a better rate for the
asymptotic regret. The corollary is valid for distributions satisfying
Assumptions 1, 2 and 3, which imply a bounded $A$: the state space can be
characterized setting $\overline{A}<\infty$ as the upper bound for $|A|$,
implying a maximum for the expectation of the asymptotic regret equal to
$\overline{A}^{\frac{2}{2h+1}}\overline{K}^{\frac{2h}{2h+1}}\underline{H}^{-1}C^{s}$.

###  3.3 Comparison of Regrets $\mathcal{R}(\hat{t}^{e}_{n})$ and
$\mathcal{R}(\hat{t}^{s}_{n})$

Corollaries 2.1 and 4.1 showed that the regret with the SWM policy has a
faster convergence rate than with the EWM policy. Assuming the accuracy of the
asymptotic approximations, the comparison can extend beyond the rates to
investigate how differently the asymptotic distributions depend on $P$,
through $H$, $K$, and $A$. Since it is more intuitive to compare expectations
than entire distributions, I will focus on this comparison, with the
understanding that similar reasoning also applies to alternative statistics of
the asymptotic distributions, such as medians or different quantiles.

The findings in the corollaries suggest that, for any sample size, one can
select distributions $P$ in a manner that makes each expectation of both the
asymptotic regrets smaller. Hence, both the EWM and the SWM can be the better
policy. If the asymptotic behavior is reflected in finite sample, this implies
that in a given application where $P$ is unknown it is impossible to determine
which policy would guarantee a smaller expected regret.

In statistical decision theory, this ambiguity is recognized: focusing on the
worst case scenario, the policy maker should choose the policy that guarantees
the smaller regret when the expectation of the asymptotic regret is maximized.
In this case, for any sample size, it means to compare
$n^{-\frac{2}{3}}\overline{K}^{\frac{2}{3}}\underline{H}^{-\frac{1}{3}}C^{e}$
and
$n^{-\frac{2h}{2h+1}}\overline{A}^{\frac{2}{2h+1}}\overline{K}^{\frac{2h}{2h+1}}\underline{H}^{-1}C^{s}$,
and choose the policy accordingly. Which policy is uniformly better does hence
depends on the application, and specifically on values of parameters
$\overline{K}$, $\underline{H}$, and $\overline{A}$, which the policy maker
must set in advance.

It is interesting to investigate for which distributions $P$ one policy is
expected to do relatively better than the other, in a pointwise sense. It’s
clear that the SWM policy does relatively better with increasing sample size
due to its faster convergence rate, for any $P$. Conversely, for any sample
size, when $h=2$, the ratio of the expectations of asymptotic regrets is
$\frac{H^{\frac{2}{3}}}{A^{\frac{2}{5}}K^{\frac{2}{15}}}\frac{C^{e}}{C^{s}}$:
the SWM policy does relatively better for larger $H$ and smaller $K$ and $A$.
The negative impact of $A$ is straightforward, since it only affects the bias
for the SWM policy, without influencing the distribution of the EWM. The role
of $H$ and $K$ is less intuitive, since they appear in both distributions but
with different exponents. It’s worth noting that the SWM policy holds a
relative advantage when the derivative of the CATE is higher, and its variance
is lower- scenarios where the benefits of selecting a threshold closer to the
optimal are larger. Further comparisons of the regrets could explore the
interplay between sample size and the distribution $P$, focusing on sequences
$P_{n}$ that change with sample size. Appendix B delves into this
investigation.

It is important to acknowledge that the relevance of these discussions on
asymptotic results relies on their ability to approximate behaviors in finite
samples, as, after all, the policymaker has only access to a finite sample of
experimental data. Guided by the theoretical results just derived, Monte Carlo
simulations in the next section allow to analyze the finite sample regrets
associated with the EWM and the SWM policies.

##  4 Monte Carlo Simulations

I examine the finite sample properties of the EWM and SWM policies using Monte
Carlo simulations. The scope of this section is twofold: first, I will provide
examples of data generating processes which lead different rankings for the
two policies in terms of asymptotic regret. Then, I will verify how the
asymptotic results approximate the finite sample distributions, and compare
the finite sample regrets of the two policies.

As data generating process, consider the following distribution $P$ of
$(Y_{0},Y_{1},D,X)$:

$\displaystyle X$ $\displaystyle\sim\mathcal{N}(0,1)$
$\displaystyle\epsilon_{1}$ $\displaystyle\sim\mathcal{N}(0,\gamma)$
$\displaystyle Y_{1}$
$\displaystyle=X^{3}+\beta_{2}X^{2}+\beta_{1}X+\epsilon_{1}$ $\displaystyle
Y_{0}$ $\displaystyle\sim\mathcal{N}(0,\gamma)$ $\displaystyle D$
$\displaystyle\sim\text{Bern}(p).$

Under $P$, the potential outcome $Y_{0}$ does not depend on the index $X$, and
the treatment is randomly assigned with constant probability $p$. Parameter
values are chosen such that $\mathbb{E}[Y_{1}|X=x]$ and hence
$\mathbb{E}[Y_{1}-Y_{0}|X=x]$ are increasing function of $x$, and the optimal
threshold $t^{*}$ is 0. It can be verified that such $P$ implies the
following:

$\displaystyle K=$
$\displaystyle\phi(0)\left(\frac{\gamma^{2}}{p}+\frac{\gamma^{2}}{1-p}\right)$
$\displaystyle H=$ $\displaystyle\phi(0)\beta_{1}$ $\displaystyle A=$
$\displaystyle-\phi(0)\beta_{2}$ $\displaystyle W(t)=$
$\displaystyle\beta_{2}\left(1-\Phi(t)+t\phi(t)\right)+\beta_{1}\phi(t)+\left(t^{2}\phi(t)+2\phi(t)\right)$

where $\phi(t)$ and $\Phi(t)$ are the pdf and the cdf of the standard normal
distribution, respectively.

I consider two models characterized by different parameter values:

Model | $\gamma$ | $\beta_{1}$ | $\beta_{2}$ | $p$
---|---|---|---|---
1 | 1 | 1 | -0.5 | 0.5
2 | 3 | 0.5 | -1 | 0.5

Table 1 reports values for $K$, $H$, and $A$, and asymptotic expected regrets
for both policies for sample of size $n\in\\{500,1000,2000,3000\\}$. For the
SWM policy, asymptotic regrets are computed using the infeasible optimal
bandwidth $\sigma_{n}^{*}$. Compared to Model 1, Model 2 entails larger $K$
and $A$, and smaller $H$. Results from the previous section hence imply the
regret with the EWM policy being relatively lower in Model 2. Consider the
case with $n=500$. In model 1, the asymptotic expected regret is higher with
the EWM policy, whereas in model 2, it’s higher with the SWM policy: this
confirms that the ranking of asymptotic expected regrets depends on the
unknown data distribution $P$. Because of the fastest rate, though, as $n$
increases the SWM policy exhibits relatively better performance. Regardless of
the specific distribution $P$, there exists a certain sample size beyond which
the asymptotic expected regret with the SWM policy becomes smaller. In model
2, when $n=1,000$, the inversion of ranking already occurs.

Table 1:  Values of $K$, $H$, and $A$, and asymptotic expected regret using both EWM and SWM policies across different models, are presented. The regret for the SWM policy is computed using the optimal bandwidth $\sigma_{n}^{*}$. To facilitate the reading, asymptotic expected regrets have been scaled by a factor of 10,000. Model | n | EWM | SWM | K | H | A
---|---|---|---|---|---|---
1 | $500$ | $96.190$ | $39.714$ | $1.596$ | $0.399$ | $0.199$
$1,000$ | $60.596$ | $22.809$ | $1.596$ | $0.399$ | $0.199$
$2,000$ | $38.173$ | $13.101$ | $1.596$ | $0.399$ | $0.199$
$3,000$ | $29.131$ | $9.471$ | $1.596$ | $0.399$ | $0.199$
2 | $500$ | $400.166$ | $439.442$ | $9.575$ | $0.199$ | $0.399$
$1,000$ | $252.089$ | $252.393$ | $9.575$ | $0.199$ | $0.399$
$2,000$ | $158.806$ | $144.962$ | $9.575$ | $0.199$ | $0.399$
$3,000$ | $121.192$ | $104.805$ | $9.575$ | $0.199$ | $0.399$

To investigate the finite sample distributions of regret, I draw samples of
size $n$ from $P$ 5,000 times for each model. Each sample is used to estimate
the thresholds $\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$. Estimating
$\hat{t}^{s}_{n}$ requires specifying a bandwidth $\sigma_{n}$, for which I
adopt the following method: I use the estimated policy $\hat{t}^{e}_{n}$ to
compute $\hat{A}_{n}$ and $\hat{K}_{n}$, which are then used to compute the
optimal $\hat{\lambda}_{n}^{*}$, and the optimal bandwidth
$\hat{\sigma}_{n}^{*}$. The SWM policy $\hat{t}^{s}_{n}$ is hence estimated
with this bandwidth $\hat{\sigma}_{n}^{*}$, which consistently estimates the
optimal one if $\hat{A}_{n}$ and $\hat{K}_{n}$ consistently estimate $A$ and
$K$. I also consider $\hat{t}^{s}_{n}$ with the infeasible optimal
$\sigma_{n}^{*}$ computed from the data generating process.

Estimates for $\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$ are used to compute
regrets $\mathcal{R}(\hat{t}^{e}_{n})$ and $\mathcal{R}(\hat{t}^{s}_{n})$. I
thus obtain the finite sample distributions of the regret, which can be
compared with the asymptotic distributions derived in Corollaries 2.1 and 4.1.
Table 2 presents the mean of these finite sample and asymptotic distributions,
also depicted in Figures 1 and 2. Corresponding tables and figures for the
median regret are provided in Appendix C.

The last column of each table reports the ratio between the finite sample mean
regrets for the EWM and SWM policy, facilitating the comparison: a ratio
larger than one indicates that the SWM policy outperforms the EWM policy.
These ratios increase with the sample size, reflecting the faster asymptotic
convergence rate of the SWM policy. Similar to the asymptotic results, in
finite sample the SWM policy does relatively better as the sample size
increases.

Table 2:  Finite sample and asymptotic expected regrets with the EWM and SWM policies across different models are presented. Finite sample (empirical) values are computed through 5,000 Monte Carlo simulations. The last column displays the ratio between finite sample expected regrets with the EWM and SWM policies. Model | n | EWM | SWM | Ratio
---|---|---|---|---
|  | empirical | asymptotic |  empirical $\sigma_{n}^{*}$ |  empirical $\hat{\sigma}_{n}^{*}$ | asymptotic |
1 | $500$ | $89.635$ | $96.190$ | $31.492$ | $77.747$ | $39.714$ | $1.153$
$1,000$ | $58.799$ | $60.596$ | $18.248$ | $44.240$ | $22.809$ | $1.329$
$2,000$ | $37.297$ | $38.173$ | $10.887$ | $29.176$ | $13.101$ | $1.278$
$3,000$ | $28.615$ | $29.131$ | $7.872$ | $16.189$ | $9.471$ | $1.768$
2 | $500$ | $276.867$ | $400.166$ | $215.876$ | $291.654$ | $439.442$ | $0.949$
$1,000$ | $201.182$ | $252.089$ | $151.181$ | $209.069$ | $252.393$ | $0.962$
$2,000$ | $149.572$ | $158.806$ | $107.733$ | $151.069$ | $144.962$ | $0.990$
$3,000$ | $124.168$ | $121.192$ | $90.251$ | $125.565$ | $104.805$ | $0.989$

Figure 1: The figure illustrates asymptotic and finite sample expected regrets
for the EWM and SWM policies in Model 1, corresponding to the values reported
in Table 2.

Figure 2: The figure illustrates asymptotic and finite sample expected regrets
for the EWM and SWM policies in Model 2, corresponding to the values reported
in Table 2.

Simulations enable comparison between finite sample regrets and their
asymptotic counterparts. Across all models and sample sizes, the asymptotic
approximation for the feasible SWM policy (with the estimated bandwidth
$\hat{\sigma}_{n}^{*}$) is relatively less accurate. Simulations suggest that
this is partly attributable to the need for estimating an additional tuning
parameter, the bandwidth $\sigma_{n}$. When the SWM policy is estimated using
the infeasible optimal bandwidth $\sigma^{*}_{n}$, in fact, the asymptotic
approximation is more accurate and the regret is smaller.

In Model 1, as illustrated in Figure 1, both the finite sample and the
asymptotic expected regrets are lower for the SWM policy. Conversely, in Model
2 (illustrated in Figure 2), the finite sample expected regret is lower with
the EWM policy. This confirms the impossibility of ranking the policies in a
pointwise sense: different distributions $P$ result in different rankings for
the finite sample expected regrets.

It is important to note that the ranking of the EWM and the SWM policies
indicated by the asymptotic results may differ from the actual finite sample
comparison. Consider, for example, Model 2 with $n=2,000$: despite the
asymptotic analysis suggesting a smaller expected regret with the SWM policy,
the EWM actually guarantees a smaller regret. In this scenario, even if $P$
were known in advance, choosing according to the asymptotic approximation
would not have been optimal. As $n$ increases, the approximation improves, and
the rankings based on asymptotic analysis and finite sample comparisons
coincide.

Monte Carlo simulations have confirmed that the asymptotic results can
approximate some finite sample behavior of the regrets, highlighting some
caveats to consider when applying conclusions from asymptotic analysis to
finite sample regrets with the EWM and SWM policies. However, they have not
yet provided insight into the practical significance of the differences
between the two policies, whether these differences are relevant or negligible
in real-world scenarios. An empirical illustration is useful to answer these
questions, illustrating the different implications that the policies may have.

##  5 Empirical Illustration

I consider the same empirical setting as kitagawa2018should: experimental data
from the National Job Training Partnership Act (JTPA) Study. bloom1997benefits
describes the experiment in detail. The study randomized whether applicants
would be eligible to receive a mix of training, job-search assistance, and
other services provided by the JTPA for a period of 18 months. Background
information on the applicants was collected before treatment assignment,
alongside administrative and survey data on the applicants’ earnings over the
subsequent 30 months.

I consider the same sample of 9,223 observations as in kitagawa2018should. The
outcome variable $Y$ represents the total individual earnings during the 30
months following program assignment, and the treatment variable $D$ is a
binary indicator denoting whether the individual was assigned to the program
(intention-to-treat). The threshold policy is implemented by considering the
individual’s earnings in the year preceding the assignment as the index $X$.
Treatment is exclusively assigned to workers with prior earnings below the
threshold, based on the expectation that program services yield a more
substantial positive effect for individuals who previously experienced lower
earnings. Experimental data are employed to determine the threshold beyond
which the treatment, on average, harms the recipients.

The bandwidth to estimate the SWM policy is chosen as in the Monte Carlo
simulation, using the EWM policy $\hat{t}^{e}_{n}$ to compute $\hat{A}_{n}$,
$\hat{K}_{n}$, the optimal $\hat{\lambda}_{n}^{*}$, and then the optimal
bandwidth $\hat{\sigma}_{n}^{*}$. Table 3 reports the threshold estimates,
including the confidence intervals constructed as discussed in Appendix A. The
threshold with the SWM policy is 700 dollars lower than with the EWM (5,924 vs
6,614 dollars), a drop of $10.5\%$. The lower threshold implies that the
treatment would target fewer workers: if the EWM policy were implemented,
$82.9\%$ of the workers in the sample would receive the program services,
compared to the $78.9\%$ with the SWM policy, resulting in a $4$ percentage
point decrease.

Table 3: Summary of the Empirical Welfare Maximizer (EWM) and Smoothed Welfare Maximizer (SWM) policies.  | EWM | SWM
---|---|---
Optimal threshold | 6614 | 5924
Confidence interval | (4880.6, 8347.4) | (5832.7, 6411.3)
Bootstrapped confidence interval | (4748.5, 8060) |
Expected asymptotic $\mathcal{R}$ | 41.299 | 5.296
Median asymptotic $\mathcal{R}$ | 19.47 | 2.597
% of workers treated | 82.912 | 78.911

Results in corollaries 2.1 and 4.1 allow to estimate the expected and the
median regret with the two policies: the asymptotic approximation suggests
that the average regret drops from $42$ to $5.5$ dollars per worker, when
comparing the EWM and SWM. In this context, the SWM would guarantee an average
gain of $37.6$ dollars per worker over the 30-month period under study,
equating to $909$ dollars on average for the workers who would change their
treatment assignment under the two policies.

The numbers in the table should be considered with care, and clearly, the
intention of this empirical illustration was not to advocate for a specific
new job-training policy. Rather, the application aimed to assess if the EWM
and SWM policies may have implications with relevant economic differences.
Results suggest this is the case: together with theoretical and simulation
findings, this implies that the choice between the EWM and the SWM policy
should be thoughtfully considered, as it may determine relevant improvement in
population welfare.

##  6 Conclusion

In this paper, I addressed the problem of using experimental data to estimate
optimal threshold policies when the policymaker seeks to minimize the regret
associated with implementing the policy in the population. I first examined
the Empirical Welfare Maximizer threshold policy, deriving its asymptotic
distribution, and showing how it links to the asymptotic distribution of its
regret. I then introduced the Smoothed Welfare Maximizer policy, replacing the
indicator function in the EWM policy with a smooth kernel function. Under the
assumptions commonly made in the policy learning literature, the convergence
rate for the worst-case regret of the SWM is faster than with the EWM policy.
A comparative analysis of the asymptotic distributions of the two policies was
conducted, to investigate how differently their regrets depend on the data
distribution $P$. Monte Carlo simulations corroborated the asymptotic finding
that the SWM policy may perform better than the commonly studied EWM policy
also in finite sample. An empirical illustration displayed that the
implications of the two policies can remarkably differ in real-world
application.

Three sets of problems remain open for future research, to extend the results
of this paper in diverse directions. While this study compared the EWM policy
with its smoothed counterpart SWM, the literature in statistical decision
theory has also examined alternative policy functions. Consider, for example,
the Augmented Inverse Propensity Weighted policy proposed by athey2021policy:
what is the asymptotic distribution of the AIPW policy’s regret in the context
of threshold policies? How differently does it depend on $P$? Is it possible,
analogously to what I did in this paper for the EWM policy, to modify the AIPW
policy by smoothing the indicator function in its definition?

Second, it would be interesting to extend the smoothing approach from the EWM
policy to other policy classes. Threshold policies are convenient as they
depend on only one parameter. Still, besides more convoluted derivations, the
same intuition of smoothing the indicator function also seems valid for the
linear index or the multiple indices policy. The questions to explore include
adapting the theory developed in this paper to these policy classes and
whether this approach could be generalized even to all cases where the EWM
policy is applied.

Lastly, the framework developed in this paper for using experimental data to
estimate optimal policies could inform experimental design. While the existing
literature mainly focuses on optimal design for estimating the average
treatment effect, it could be valuable to consider scenarios when estimating
the threshold policy is the goal: how should the experimental design be
adapted? How the allocation of units to treatment and control groups would
change? The results presented in this paper, elucidating the connection
between the distribution $P$ and the regret of the policy, provide a natural
foundation for exploring experimental designs optimal for threshold policy
estimation.

## References

  * Abrevaya and Huang (2005) Abrevaya, J. and J. Huang (2005).  On the bootstrap of the maximum score estimator.  Econometrica 73(4), 1175–1204.
  * Aiken et al. (2022) Aiken, E., S. Bellue, D. Karlan, C. Udry, and J. E. Blumenstock (2022).  Machine learning and phone data can improve targeting of humanitarian aid.  Nature 603(7903), 864–870.
  * Amemiya (1985) Amemiya, T. (1985).  Advanced econometrics.  Harvard university press.
  * Athey and Wager (2021) Athey, S. and S. Wager (2021).  Policy learning with observational data.  Econometrica 89(1), 133–161.
  * Banerjee and McKeague (2007) Banerjee, M. and I. W. McKeague (2007).  Confidence sets for split points in decision trees.  The Annals of Statistics 35(2), 543–574.
  * Banerjee and Wellner (2001) Banerjee, M. and J. A. Wellner (2001).  Likelihood ratio tests for monotone functions.  Annals of Statistics, 1699–1731.
  * Bloom et al. (1997) Bloom, H. S., L. L. Orr, S. H. Bell, G. Cave, F. Doolittle, W. Lin, and J. M. Bos (1997).  The benefits and costs of jtpa title ii-a programs: Key findings from the national job training partnership act study.  Journal of human resources, 549–576.
  * Card et al. (2008) Card, D., C. Dobkin, and N. Maestas (2008).  The impact of nearly universal insurance coverage on health care utilization: evidence from medicare.  American Economic Review 98(5), 2242–2258.
  * Cattaneo et al. (2020) Cattaneo, M. D., M. Jansson, and K. Nagasawa (2020).  Bootstrap-based inference for cube root asymptotics.  Econometrica 88(5), 2203–2219.
  * Chernoff (1964) Chernoff, H. (1964).  Estimation of the mode.  Annals of the Institute of Statistical Mathematics 16(1), 31–41.
  * Crost et al. (2014) Crost, B., J. Felter, and P. Johnston (2014).  Aid under fire: Development projects and civil conflict.  American Economic Review 104(6), 1833–1856.
  * Groeneboom and Wellner (2001) Groeneboom, P. and J. A. Wellner (2001).  Computing chernoff’s distribution.  Journal of Computational and Graphical Statistics 10(2), 388–400.
  * Haushofer et al. (2022) Haushofer, J., P. Niehaus, C. Paramo, E. Miguel, and M. W. Walker (2022).  Targeting impact versus deprivation.  Technical report, National Bureau of Economic Research.
  * Hirano and Porter (2009) Hirano, K. and J. R. Porter (2009).  Asymptotics for statistical treatment rules.  Econometrica 77(5), 1683–1701.
  * Horowitz (1992) Horowitz, J. L. (1992).  A smoothed maximum score estimator for the binary response model.  Econometrica: journal of the Econometric Society, 505–531.
  * Hussam et al. (2022) Hussam, R., N. Rigol, and B. N. Roth (2022).  Targeting high ability entrepreneurs using community information: Mechanism design in the field.  American Economic Review 112(3), 861–98.
  * Kamath and Kim (2007) Kamath, P. S. and W. R. Kim (2007).  The model for end-stage liver disease (meld).  Hepatology 45(3), 797–805.
  * Kim and Pollard (1990) Kim, J. and D. Pollard (1990).  Cube root asymptotics.  The Annals of Statistics, 191–219.
  * Kitagawa et al. (2022) Kitagawa, T., S. Lee, and C. Qiu (2022).  Treatment choice with nonlinear regret.  arXiv preprint arXiv:2205.08586.
  * Kitagawa and Tetenov (2018) Kitagawa, T. and A. Tetenov (2018).  Who should be treated? empirical welfare maximization methods for treatment choice.  Econometrica 86(2), 591–616.
  * Leboeuf et al. (2020) Leboeuf, J.-S., F. LeBlanc, and M. Marchand (2020).  Decision trees as partitioning machines to characterize their generalization properties.  Advances in Neural Information Processing Systems 33, 18135–18145.
  * Léger and MacGibbon (2006) Léger, C. and B. MacGibbon (2006).  On the bootstrap in cube root asymptotics.  Canadian Journal of Statistics 34(1), 29–44.
  * Manski (1975) Manski, C. F. (1975).  Maximum score estimation of the stochastic utility model of choice.  Journal of econometrics 3(3), 205–228.
  * Manski (2004) Manski, C. F. (2004).  Statistical treatment rules for heterogeneous populations.  Econometrica 72(4), 1221–1246.
  * Manski (2021) Manski, C. F. (2021).  Econometrics for decision making: Building foundations sketched by haavelmo and wald.  Econometrica 89(6), 2827–2853.
  * Manski (2023) Manski, C. F. (2023).  Probabilistic prediction for binary treatment choice: with focus on personalized medicine.  Journal of Econometrics 234(2), 647–663.
  * Manski and Tetenov (2023) Manski, C. F. and A. Tetenov (2023).  Statistical decision theory respecting stochastic dominance.  The Japanese Economic Review 74(4), 447–469.
  * Mbakop and Tabord-Meehan (2021) Mbakop, E. and M. Tabord-Meehan (2021).  Model selection for treatment choice: Penalized welfare maximization.  Econometrica 89(2), 825–848.
  * Mohammadi et al. (2005) Mohammadi, L., S. Van De Geer, and J. Shawe-Taylor (2005).  Asymptotics in empirical risk minimization.  Journal of Machine Learning Research 6(12).
  * Rai (2018) Rai, Y. (2018).  Statistical inference for treatment assignment policies.  Unpublished Manuscript.
  * Shigeoka (2014) Shigeoka, H. (2014).  The effect of patient cost sharing on utilization, health, and risk protection.  American Economic Review 104(7), 2152–2184.
  * Stoye (2012) Stoye, J. (2012).  Minimax regret treatment choice with covariates or with limited validity of experiments.  Journal of Econometrics 166(1), 138–156.
  * Sun et al. (2021) Sun, H., E. Munro, G. Kalashnov, S. Du, and S. Wager (2021).  Treatment allocation under uncertain costs.  arXiv preprint arXiv:2103.11066.
  * Sun (2021) Sun, L. (2021).  Empirical welfare maximization with constraints.  arXiv preprint arXiv:2103.15298.
  * Taylor (2003) Taylor, J. (2003).  Corporation income tax brackets and rates, 1909-2002.  Statistics of Income. SOI Bulletin 23(2), 284–291.
  * Viviano and Bradic (2023) Viviano, D. and J. Bradic (2023).  Fair policy targeting.  Journal of the American Statistical Association, 1–14.

##  Appendix A Confidence Intervals for Threshold Policies

Results derived in Section 3 can be used to construct confidence intervals
that asymptotically cover the optimal threshold policy with a given
probability, and to conduct hypotheses tests. It is important to remark that,
in a decision problem setting, hypotheses testing does not have a clearly
motivated justification, and indeed, statistical decision theory is the
alternative approach to deal with decisions under uncertainty, as pointed out
in manski2021econometrics. Rather than advocating for confidence intervals and
hypothesis tests for threshold policies, this appendix aims to provide a
procedure agnostic on why one may be interested in it.

For the EWM policy, rai2018statistical proposes some confidence intervals
uniformly valid for several policy classes. They rely on test inversion of a
certain bootstrap procedure, which compares the welfare generated by all the
policies in the class. My procedure is much simpler for the EWM threshold
policies, and I directly construct confidence intervals from the asymptotic
distributions derived in Theorem 2. An analogous approach, built over results
in Theorem 4, is then used to construct confidence intervals for the SWM
policy.

###  A.1 Empirical Welfare Maximizer Policy

Consider the asymptotic distribution for the EWM threshold policy derived in
Theorem 2:

$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right).$ (16)

If $H$ and $K$ were known, confidence intervals for the optimal policy $t^{*}$
with asymptotic coverage $1-\alpha$ could be constructed as
$(\hat{t}^{e}_{n}-w^{e}_{n},\hat{t}^{e}_{n}+w^{e}_{n})$, where

$\displaystyle w^{e}_{n}=n^{-1/3}(2\sqrt{K}/H)^{2/3}c_{\alpha/2}$ (17)

and $c_{\alpha/2}$ is the critical value, the upper $\alpha/2$ quantile of the
distribution of $\max_{r}B(r)-r^{2}$.

In practice, $H$ and $K$ are unknown and should be estimated. They are defined
as:

$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
$\displaystyle H=$ $\displaystyle
f_{x}(t^{*})\left(\frac{\partial\mathbb{E}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$

and can be estimated by a plug-in method: consider kernel density estimator
$\hat{f}_{x}(x)$ for $f_{x}(x)$, and local linear estimators
$\hat{\kappa}_{j}(x)$ and $\hat{\nu}_{j}^{\prime}(x)$ for
$\kappa_{j}(x)=\mathbb{E}\left[Y_{j}^{2}|X=x,D=j\right]$ and
$\nu_{j}^{\prime}(x)=\frac{\partial\nu_{j}(x)}{\partial
x}=\frac{\partial\mathbb{E}\left[Y_{j}|X=x,D=j\right]}{\partial x}$. Define
estimators $\hat{K}_{n}$ and $\hat{H}_{n}$ by:

$\displaystyle\hat{K}_{n}=\hat{f}_{x}(\hat{t}^{e}_{n})\left(\frac{1}{p(\textbf{X})}\hat{\kappa}_{1}(\hat{t}^{e}_{n})+\frac{1}{1-p(\textbf{X})}\hat{\kappa}_{0}(\hat{t}^{e}_{n})\right)$
(18)

and

$\displaystyle\hat{H}_{n}=\hat{f}_{x}(\hat{t}^{e}_{n})(\hat{\nu}_{1}^{\prime}(\hat{t}^{e}_{n})-\hat{\nu}_{0}^{\prime}(\hat{t}^{e}_{n})).$
(19)

Under the additional assumption that the second derivatives of $f_{x}$,
$\nu_{1}$ and $\nu_{0}$ are continuous and bounded in a neighborhood of
$t^{*}$, and with the proper choice of bandwidth sequences, $\hat{K}_{n}$ and
$\hat{H}_{n}$ are consistent estimators for $K$ and $H$.

Feasible confidence intervals with asymptotic coverage $1-\alpha$ can hence be
constructed as
$(\hat{t}^{e}_{n}-\hat{w}^{e}_{n},\hat{t}^{e}_{n}+\hat{w}^{e}_{n})$, where

$\displaystyle\hat{w}^{e}_{n}=n^{-1/3}(2\sqrt{\hat{K}_{n}}/\hat{H}_{n})^{2/3}c_{\alpha/2}.$
(20)

####  A.1.1 Bootstrap

To avoid relying on tabulated values for $c_{\alpha/2}$ and on estimation of
$K$, an alternative approach to inference for the EWM policy is the bootstrap.
Nonparametric bootstrap is not valid for $\hat{t}^{e}_{n}$ and, more
generally, for “cube root asymptotics” estimators (abrevaya2005bootstrap;
leger2006bootstrap). Nonetheless, cattaneo2020bootstrap provide a consistent
bootstrap procedure for estimators of this type. Consistency is achieved by
altering the shape of the criterion function defining the estimator whose
distribution must be approximated. The standard nonparametric bootstrap is
inconsistent for $Q_{0}(r)=-\frac{1}{2}Hr$ as defined in the proof of Theorem
2, and hence the procedure in cattaneo2020bootstrap directly estimates this
non-random part.

Let $\\{Z_{i}^{b}\\}$ be a random sample from the empirical distribution
$P_{n}$, and define the estimator $\hat{t}^{b}_{n}$ as:

$\displaystyle\hat{t}^{b}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}^{b}Y_{i}^{b}}{p(\textbf{X}_{i}^{b})}-\frac{(1-D_{i}^{b})Y_{i}^{b}}{(1-p(\textbf{X}_{i}^{b}))}\right)\mathbf{1}\\{X_{i}^{b}>t\\}\right]-$
(21)
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)\mathbf{1}\\{X_{i}>t\\}\right]-\frac{1}{2}(t-\hat{t}_{n})^{2}\hat{H}_{n}.$
(22)

The bootstrap procedure proposed by cattaneo2020bootstrap is the following:

  1. 1.

Compute $\hat{t}^{e}_{n}$ as described in equation (9).

  2. 2.

Using $\hat{t}^{e}_{n}$, compute $\hat{H}_{n}$ as described in equation (19).

  3. 3.

Using $\hat{t}^{e}_{n}$, $\hat{H}_{n}$, and the bootstrap sample
$\\{Z_{i}^{b}\\}$, compute $\hat{t}^{b}_{n}$ as described in equation (21).

  4. 4.

Iterate step 3 to obtain the distribution of
$n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}^{e}_{n}\right)$, and use it as an
estimate for the distribution of $n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)$.

To be valid, the procedure needs an additional assumption.

######  Assumption 6.

(Bounded 4th moment) Potential outcomes distribution are such that
$\frac{1}{n^{2/3}}\mathbb{E}[Y_{1}^{4}|X=t^{*}]=o(1)$ and
$\frac{1}{n^{2/3}}\mathbb{E}[Y_{0}^{4}|X=t^{*}]=o(1)$.

Assumption 6 guarantees that the envelope $G_{R}$ is such that
$PG_{R}^{4}=o(R^{-1})$. Theorem 5 proves that the distribution of
$n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}^{e}_{n}\right)$ consistently estimates
the distribution of $n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)$, and validate
the bootstrap procedure.

######  Theorem 5.

Consider estimators $\hat{t}^{e}_{n}$ defined in equation (9) and
$\hat{t}^{b}_{n}$ defined in equation (21) and the estimand $t^{*}$ defined in
equation (4). Under Assumptions 1, 2 with $s=2$, 3, and 6, as
$\hat{H}_{n}\rightarrow^{p}H$ and $n\rightarrow\infty$,

$\displaystyle
n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}_{n}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right)$ (23)

where the limiting distribution is the same as in Theorem 2.

Distribution of $n^{1/3}\left(\hat{t}^{b}_{n}-\hat{t}^{e}_{n}\right)$ can
hence be used to construct asymptotic valid confidence intervals and run
hypothesis tests for $\hat{t}^{e}_{n}$.

###  A.2 Smoothed Welfare Maximizer Policy

Consider the asymptotic distribution for the SWM threshold policy derived in
Theorem 4, for $n\sigma_{n}^{2h+1}\rightarrow\lambda<\infty$:

$\displaystyle(n\sigma_{n})^{1/2}(\hat{t}^{s}_{n}-t^{*})\rightarrow^{d}\mathcal{N}(\lambda^{1/2}H^{-1}A,H^{-2}\alpha_{2}K).$
(24)

$\lambda$, $\sigma_{n}$, and $\alpha_{2}$ are known. If also $K$, $H$, and $A$
were known, confidence intervals for the optimal policy $t^{*}$ with
asymptotic coverage $1-\alpha$ could be constructed as
$(\hat{t}^{s}_{n}-b_{n}-w^{s}_{n},\hat{t}^{s}_{n}-b_{n}+w^{s}_{n})$, where

$\displaystyle b_{n}=(n\sigma_{n})^{-1/2}\lambda^{1/2}\frac{A}{H}$ (25)
$\displaystyle
w^{s}_{n}=(n\sigma_{n})^{-1/2}(\sqrt{\alpha_{2}K}/H)c_{\alpha/2}$ (26)

and $c_{\alpha/2}$ the upper $\alpha/2$ quantile of the standard normal
distribution.

In practice, $K$, $H$, and $A$ are unknown and should be estimated. As usual
with inference involving bandwidths and kernels, two approaches are available:
estimate and remove the asymptotic bias, or undersmooth.

For the first approach, consider estimators in equation (18) for $\hat{K}_{n}$
and in equation (19) for $\hat{H}_{n}$, substituting $\hat{t}^{e}_{n}$ with
$\hat{t}^{s}_{n}$. For $A$, recall that

$\displaystyle A=$
$\displaystyle-\frac{1}{h!}\alpha_{1}\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{x}(y,t^{*})dy$
(27) $\displaystyle=$
$\displaystyle-\frac{1}{h!}\alpha_{1}\left[2f^{\prime}_{x}(t^{*})[\lambda^{\prime}_{1}(t^{*})-\lambda^{\prime}_{0}(t^{*})]+f_{x}(t^{*})[\lambda^{\prime\prime}_{1}(t^{*})-\lambda^{\prime\prime}_{0}(t^{*})]\right]$
(28)

where $f_{x}(x)$ is the probability density function of $X$ and
$\nu_{j}(x)=\mathbb{E}[Y_{j}|X=x,D=j]$. Consider kernel density estimator
$\hat{f}_{x}(x)$ and $\hat{f}^{\prime}_{x}(x)$ for $f_{x}(x)$ and
$f^{\prime}_{x}(x)$, and local linear estimators $\hat{\nu}_{j}^{\prime}(x)$
and $\hat{\nu}_{j}^{\prime\prime}(x)$ for
$\nu_{j}^{\prime}(x)=\frac{\partial\nu_{j}(x)}{\partial x}$ and
$\nu_{j}^{\prime\prime}(x)=\frac{\partial^{2}\nu_{j}(x)}{\partial^{2}x}$.
Define estimator $\hat{A}_{n}$ by:

$\displaystyle\hat{A}_{n}=-\frac{1}{h!}\alpha_{1}\left[2\hat{f}^{\prime}_{x}(\hat{t}^{s}_{n})[\hat{\lambda}^{\prime}_{1}(\hat{t}^{s}_{n})-\hat{\lambda}^{\prime}_{0}(\hat{t}^{s}_{n})]+\hat{f}_{x}(\hat{t}^{s}_{n})[\hat{\lambda}^{\prime\prime}_{1}(\hat{t}^{s}_{n})-\hat{\lambda}^{\prime\prime}_{0}(\hat{t}^{s}_{n})]\right]$
(29)

which consistently estimate $A$ the additional assumption that the third
derivatives of $f_{x}$, $\nu_{1}$ and $\nu_{0}$ are continuous and bounded in
a neighborhood of $t^{*}$, and with the proper choice of bandwidth sequences.

Confidence intervals with asymptotic coverage $1-\alpha$ can hence be
constructed as
$(\hat{t}^{s}_{n}-\hat{b}_{n}-\hat{w}^{s}_{n},\hat{t}^{s}_{n}-\hat{b}_{n}+\hat{w}^{s}_{n})$,
where

$\displaystyle\hat{b}_{n}=(n\sigma_{n})^{-1/2}\lambda^{1/2}\frac{\hat{A}_{n}}{\hat{H}_{n}}$
(30)
$\displaystyle\hat{w}^{s}_{n}=(n\sigma_{n})^{-1/2}(\sqrt{\alpha_{2}\hat{K}_{n}}/\hat{H}_{n})c_{\alpha/2}$
(31)

The second approach relies on undersmoothing, and chooses a suboptimally small
$\sigma_{n}$ to eliminate the asymptotic bias, with no need to estimate $A$.
Instead of a bandwidth sequence $\sigma_{n}=O_{p}(n^{-\frac{1}{2h+1}})$, it
considers a sequence $\sigma_{n}=o_{p}(n^{-\frac{1}{2h+1}})$ such that
$n\sigma_{n}^{2h+1}\rightarrow\lambda=0$, and ensures $b_{n}\to 0$. Confidence
intervals with asymptotic coverage $1-\alpha$ can hence be constructed as
$(\hat{t}^{s}_{n}-\hat{w}^{s}_{n},\hat{t}^{s}_{n}+\hat{w}^{s}_{n})$, with
$\hat{w}^{s}_{n}$ defined as above.

##  Appendix B Local Asymptotics

The interplay between the population distribution $P$ and the sample size can
be studied considering a local asymptotic framework, considering a sequence of
population distributions $\\{P_{n}\\}$ that varies with $n$. I focus on the
following two sequences.

######  Definition 1.

(Sequence $\\{P^{1}_{n}\\}$) The sequence of distributions $\\{P^{1}_{n}\\}$
is such that $\frac{\sqrt{K_{n}}}{H_{n}}=n^{\gamma}$, with
$\gamma\in[0,\frac{h}{2h+1})$, and $\frac{A_{n}}{H_{n}}=1$.

######  Definition 2.

(Sequence $\\{P^{2}_{n}\\}$) The sequence of distributions $\\{P^{2}_{n}\\}$
is such that $\frac{\sqrt{K_{n}}}{H_{n}}=1$, and
$\frac{A_{n}}{H_{n}}=n^{\gamma}$, with $\gamma\in[0,\frac{h}{2h+1})$.

Sequence $\\{P^{1}_{n}\\}$ mimics a scenario where $\sqrt{K}$ is large
compared to $H$. This occurs when the variance of the conditional treatment
effect, $\operatorname{\mathrm{Var}}(Y_{1}-Y_{0}|X=t^{*})$, is large compared
to the derivative of the conditional ATE,
$\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial X}$,
or compared to the density of the index, $f_{x}(t^{*})$. In these situations,
the population welfare remains relatively stable for thresholds in the
neighborhood of the optimal one. The limit case with $\gamma=\frac{1}{2}$
coincides with the local asymptotic framework proposed by
hirano2009asymptotics and studied by athey2021policy.

Sequence $\\{P^{2}_{n}\\}$, instead, mimics a situation where $A$ is large
compared to $H$. When $h=2$, this happens when the second derivative of the
conditional ATE,
$\frac{\partial^{2}\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial^{2}X}$,
is large compared to the first derivative,
$\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial X}$.
Since the asymptotic bias of $\hat{t}^{s}_{n}$ is
$\lambda\frac{A_{n}}{H_{n}}$, sequence $\\{P^{2}_{n}\\}$ represents situation
where this bias is large relatively to the sample size.

Let $r_{n}^{e}$ and $r_{n}^{s}$ denote the rates of convergence of
$\mathcal{R}(\hat{t}^{e}_{n})$ and $\mathcal{R}(\hat{t}^{s}_{n})$, i.e. let
$r_{n}^{e}$ and $r_{n}^{s}$ be sequences such that
$\mathcal{R}(\hat{t}^{e}_{n})=O_{p}(r_{n}^{e})$ and
$\mathcal{R}(\hat{t}^{s}_{n})=O_{p}(r_{n}^{s})$. The following theorem
establishes a relationship between $r_{n}^{e}$ and $r_{n}^{s}$ under
$\\{P^{1}_{n}\\}$ and $\\{P^{2}_{n}\\}$.

######  Theorem 6.

Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, 3, and 5 hold. Consider
sequences $\\{P^{1}_{n}\\}$ and $\\{P^{2}_{n}\\}$ of data generating
processes. The limit of the ratio $\frac{r_{n}^{e}}{r_{n}^{s}}$ depends on
$\gamma$ as follows:

  * •

$\frac{r_{n}^{e}}{r_{n}^{s}}\to\infty$ if
$\gamma\in(\bar{\gamma},\frac{h}{2h+1})$

  * •

$\frac{r_{n}^{e}}{r_{n}^{s}}=O_{p}(1)$ if $\gamma=\bar{\gamma}$

  * •

$\frac{r_{n}^{e}}{r_{n}^{s}}\to 0$ if $\gamma\in[0,\bar{\gamma})$.

where $\bar{\gamma}=\frac{h-1}{2h+1}$ under $\\{P^{1}_{n}\\}$, and
$\bar{\gamma}=\frac{1}{3}\frac{h-1}{2h+1}$ under $\\{P^{2}_{n}\\}$.

Theorem 6 shows how, under sequences $\\{P^{1}_{n}\\}$ and $\\{P^{2}_{n}\\}$,
the EWM policy guarantees a regret convergence rate faster than the SWM policy
whenever the parameter $\gamma$ exceeds a certain value $\bar{\gamma}$. For
sequence $\\{P^{1}_{n}\\}$, the result depends on the fact that the term
$\frac{\sqrt{K_{n}}}{H_{n}}$ enters the asymptotic distributions of
$\hat{t}^{e}_{n}$ and $\hat{t}^{s}_{n}$ with different powers ($\frac{2}{3}$
and 1, respectively): if $\frac{\sqrt{K_{n}}}{H_{n}}$ is large enough
relatively to the sample size, the exponent lower than one makes $r^{e}_{n}$
faster (and hence in the limit $\hat{t}^{e}_{n}$ preferable). For sequence
$\\{P^{2}_{n}\\}$, the result is due to the asymptotic bias of the SWM policy,
proportional to $\frac{A_{n}}{H_{n}}$. Since the asymptotic distribution of
the regret of the EWM policy remains constant in $\\{P^{2}_{n}\\}$, when the
bias of $\hat{t}^{s}_{n}$ is large enough compared to sample size,
$\hat{t}^{e}_{n}$ becomes preferable. The value of $\bar{\gamma}$ is
increasing in $h$, the order of the kernel $k$: a smoother objective function
$\mathbb{E}[(Y_{1}-Y_{0})\mathbf{1}\\{X>t\\}]$ amplifies the benefit of using
the SWM policy, expanding the region of values of $\gamma$ where the SWM
policy has a faster rate compared to the EWM.

Considering the sequence $\\{P^{1}_{n}\\}$ with $\gamma=\frac{1}{2}$,
athey2021policy show that their AIPW policy achieves the uniform fastest
asymptotic convergence rate. Results in Theorem 6 are different: without
focusing on a single specific sequence, their goal is to shed light on how the
asymptotic behavior of the EWM and the SWM policies depends on $P$ and sample
size.

##  Appendix C Monte Carlo Simulation

I report the analogous of Table 2 and Figures 1 and 2 for the median regret.
Comments made for the mean also extend to the median regret.

Table 4:  Finite sample and asymptotic median regret with the EWM and SWM policies across different models. Finite sample (empirical) values are computed through 5,000 Monte Carlo simulations. The last column reports the ratio between finite sample median regrets with the EWM and SWM policies. Model | n | EWM | SWM | Ratio
---|---|---|---|---
|  | empirical | asymptotic |  empirical $\sigma_{n}^{*}$ |  empirical $\hat{\sigma}_{n}^{*}$ | asymptotic |
1 | $500$ | $39.809$ | $45.347$ | $15.113$ | $24.169$ | $18.459$ | $1.647$
$1,000$ | $27.547$ | $28.567$ | $8.692$ | $14.145$ | $10.602$ | $1.948$
$2,000$ | $18.816$ | $17.996$ | $5.073$ | $8.507$ | $6.089$ | $2.212$
$3,000$ | $14.043$ | $13.733$ | $3.471$ | $5.491$ | $4.402$ | $2.558$
2 | $500$ | $147.029$ | $188.651$ | $116.784$ | $159.876$ | $204.255$ | $0.920$
$1,000$ | $110.372$ | $118.843$ | $82.473$ | $118.695$ | $117.314$ | $0.930$
$2,000$ | $87.325$ | $74.866$ | $55.731$ | $79.436$ | $67.379$ | $1.099$
$3,000$ | $72.122$ | $57.134$ | $46.007$ | $68.656$ | $48.714$ | $1.050$

Figure 3: Figure reports asymptotic and finite sample median regrets for the
EWM and SWM policies, illustrating the values reported in Table 4.

Figure 4: Figure reports asymptotic and finite sample median regrets for the
EWM and SWM policies, illustrating the values reported in Table 4.

##  Appendix D Proofs

### Theorem 1

######  Theorem 1.

Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1 and 2 with
$s=0$,

$\displaystyle\hat{t}^{e}_{n}\rightarrow^{a.s.}t^{*}$

i.e. $\hat{t}^{e}_{n}$ is a consistent estimator for $t^{*}$.

###### Proof.

Estimator $\hat{t}^{e}_{n}$ in (9) can be written as

$\displaystyle\hat{t}^{e}_{n}=$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)(\mathbf{1}\\{X_{i}>t\\}-\mathbf{1}\\{X_{i}>t^{*}\\})\right]$
(32) $\displaystyle=$ $\displaystyle\mathop{\rm
arg~{}max}\limits_{t}\frac{1}{n}\sum_{i=1}^{n}m\left(Z_{i},t\right)$ (33)

where

$\displaystyle
m\left(Z,t\right)=\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\left(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\}\right)$
(34)

and $\\{Z_{i}\\}$ is a sample of $n$ observation from distribution $P$.

Define $P_{n}m\left(\cdot,t\right)=\sum_{i=1}^{n}m\left(Z_{i},t\right)$ and
$Pm\left(Z,t\right)=\mathbb{E}_{P}[m\left(Z,t\right)]$. With this notation,
equation (4) and (9) can be written as

$\displaystyle t^{*}=\mathop{\rm arg~{}max}\limits_{t}Pm\left(Z,t\right)$ (35)
$\displaystyle\hat{t}^{e}_{n}=\mathop{\rm
arg~{}max}\limits_{t}P_{n}m\left(\cdot,t\right).$ (36)

Threshold policies I am considering can be seen as a tree partition of depth
1. Tree partitions of finite depth are a VC class (leboeuf2020decision), and
hence $m\left(\cdot,t\right)$ is a manageable class of functions. Consider the
envelope function
$F=2\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|$.
Note that
$\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|^{2}\right]=\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}]$:
Assumption 2.2 guarantees the existence of
$\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|^{2}\right]$.

It follows from corollary 3.2 in kim1990cube that

$\displaystyle\sup_{t}\left|P_{n}m(\cdot,t)-Pm(Z,t)\right|\rightarrow^{a.s.}0.$
(37)

Under Assumptions 2.1 and 2.3, $Pm(Z,t)$ is continuous in $t$, and $t^{*}$ is
the unique maximizer. Hence,

$\displaystyle\sup_{t}\left|P_{n}m(\cdot,t)-Pm(Z,t)\right|+Pm(Z,t^{*})\geq$
(38)
$\displaystyle\left|P_{n}m(\cdot,\hat{t}^{e}_{n})-Pm(Z,\hat{t}^{e}_{n})\right|+Pm(Z,t^{*})\geq$
(39)
$\displaystyle\left|P_{n}m(\cdot,\hat{t}^{e}_{n})-Pm(Z,\hat{t}^{e}_{n})\right|+Pm(Z,\hat{t}^{e}_{n})\geq$
(40) $\displaystyle P_{n}m(\cdot,\hat{t}^{e}_{n})\geq
P_{n}m(\cdot,t^{*})\rightarrow Pm(Z,t^{*})$ (41)

where the second inequality is due to the fact that $t^{*}$ is the maximizer
of $Pm(Z,t)$, the third comes from the triangular inequality, the fourth from
$\hat{t}^{e}_{n}$ being the maximizer of $P_{n}m(\cdot,t)$, and the last limit
comes from LLN. This prove that
$P_{n}m(\cdot,\hat{t}^{e}_{n})\rightarrow^{a.s.}Pm(Z,t^{*})$, and hence
$Pm(Z,\hat{t}^{e}_{n})\rightarrow^{a.s.}Pm(Z,t^{*})$. Since $t^{*}$ is the
unique maximizer of $Pm(Z,t)$ and $Pm(Z,t)$ is continuous,
$\hat{t}^{e}_{n}\rightarrow^{a.s.}t^{*}$. It means that $\hat{t}^{e}_{n}$ is a
consistent estimator for $t^{*}$. ∎

### Theorem 2

######  Theorem 2.

Consider the EWM policy $\hat{t}^{e}_{n}$ defined in equation (9) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=2$ and 3, as $n\rightarrow\infty$,

$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}\left(B(r)-r^{2}\right)$ (42)

where $B(r)$ is the two-sided Brownian motion process, and $K$ and $H$ are

$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
$\displaystyle H=$ $\displaystyle
f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$

###### Proof.

The proof shows that conditions for the main theorem in kim1990cube hold;
hence, their result is valid for $\hat{t}^{e}_{n}$. For completeness, I report
the theorem.

######  Theorem Kim and Pollard.

Consider estimators defined by maximization of processes

$\displaystyle
P_{n}g(\cdot,\theta)=\frac{1}{n}\sum_{i=1}^{n}g\left(\xi_{i},\theta\right)$
(43)

where $\left\\{\xi_{i}\right\\}_{i}$ is a sequence of i.i.d. observations from
a distribution $P$ and $\\{g(\cdot,\theta):\theta\in\Theta\\}$ is a class of
functions indexed by a subset $\Theta$ in $\mathbb{R}^{k}$. The envelope
$G_{R}(\cdot)$ is defined as the supremum of $g(\cdot,\theta)$ over the class

$\displaystyle\mathcal{G}_{R}=\left\\{|g(\cdot,\theta)|:\left|\theta-\theta_{0}\right|\leq
R\right\\},\quad R>0.$ (44)

Let $\left\\{\theta_{n}\right\\}$ be a sequence of estimators for which

  1. 1.

$P_{n}g\left(\cdot,\theta_{n}\right)\geq\sup_{\theta\in\Theta}P_{n}g(\cdot,\theta)-o_{P}\left(n^{-2/3}\right)$.

  2. 2.

$\theta_{n}$ converges in probability to the unique $\theta_{0}$ that
maximizes $Pg(\cdot,\theta)$.

  3. 3.

$\theta_{0}$ is an interior point of $\Theta$.

Let the functions be standardized so that
$g\left(\cdot,\theta_{0}\right)\equiv 0$. If the classes $\mathcal{G}_{R}$ for
$R$ near 0 are uniformly manageable for the envelopes $G_{R}$ and satisfy:

  1. 4.

$Pg(\cdot,\theta)$ is twice differentiable with second derivative matrix $-H$
at $\theta_{0}$.

  2. 5.

$K(s,r)=\lim_{\alpha\rightarrow\infty}\alpha
Pg\left(\cdot,\theta_{0}+r/\alpha\right)g\left(\cdot,\theta_{0}+s/\alpha\right)$
exists for each $s,r$ in $\mathbb{R}^{k}$ and
$\lim_{\alpha\rightarrow\infty}\alpha
Pg\left(\cdot,\theta_{0}+r/\alpha\right)^{2}\left\\{\left|g\left(\cdot,\theta_{0}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}=0$
for each $\varepsilon>0$ and $r$ in $\mathbb{R}^{k}$.

  3. 6.

$PG_{R}^{2}=O(R)$ as $R\rightarrow 0$ and for each $\varepsilon>0$ there is a
constant C such that $PG_{R}^{2}\mathbf{1}\\{G_{R}>C\\}\leq\varepsilon R$ for
$R$ near 0.

  4. 7.

$P\left|g\left(\cdot,\theta_{1}\right)-g\left(\cdot,\theta_{2}\right)\right|=O\left(\left|\theta_{1}-\theta_{2}\right|\right)$
near $\theta_{0}$.

Then, the process $n^{2/3}P_{n}g\left(\cdot,\theta_{0}+rn^{-1/3}\right)$
converges in distribution to a Gaussian process $Q(r)$ with continuous sample
paths, expected value $-\frac{1}{2}r^{\prime}Hr$ and covariance kernel $K$. If
$H$ is positive definite and if $Q$ has nondegenerate increments, then
$n^{1/3}\left(\theta_{n}-\theta_{0}\right)$ converges in distribution to the
(almost surely unique) random vector that maximizes $Q$.

I apply Theorem Theorem Kim and Pollard by taking
$\xi_{i}=Z_{i},\theta=t,\theta_{n}=\hat{t}^{e}_{n},\theta_{0}=t^{*}$,
$g(\cdot,\theta)=m(\cdot,t)$, where $m(\cdot,t)$ is standardized:

$\displaystyle
m\left(Z_{i},t\right)=\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)\left(\mathbf{1}\\{X_{i}>t\\}-\mathbf{1}\\{X_{i}>t^{*}\\}\right).$
(45)

First, I will verify that conditions 1-7 apply to my setting:

  1. 1.

$P_{n}m(\cdot,t)$ takes only finite ($n+1$) values; hence, condition 1 is
satisfied with the equality.

  2. 2.

In Theorem 1, I proved that $\hat{t}^{e}_{n}$ is a consistent estimator for
$t^{*}$.

  3. 3.

$t^{*}$ is an interior point of $\mathcal{T}$ by Assumption 2.1.

I need to prove that the classes $\mathcal{G}_{R}$ for $R$ near 0 are
uniformly manageable for the envelopes $G_{R}$. The envelope function
$G_{R}(\cdot)$ is defined as

$\displaystyle G_{R}(z)$
$\displaystyle=\sup\left\\{m(z,t):\left|t-t^{*}\right|\leq R\right\\}$ (46)
$\displaystyle=\sup_{\left|t-t^{*}\right|\leq
R}\left[\left(\frac{dy}{p(\textbf{x})}-\frac{(1-d)y}{(1-p(\textbf{x}))}\right)\left(\mathbf{1}\\{x>t\\}-\mathbf{1}\\{x>t^{*}\\}\right)\right]$
(47)
$\displaystyle=\left|\frac{dy}{p(\textbf{x})}-\frac{(1-d)y}{(1-p(\textbf{x}))}\right|\mathbf{1}\\{\left|x-t^{*}\right|<R\\}$
(48)

and I have:

$\displaystyle PG_{R}^{2}$
$\displaystyle=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\\{\left|X-t^{*}\right|<R\\}\right]$
(49)
$\displaystyle=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Bigg{|}X\in(t^{*}-R,t^{*}+R)\right]$
(50)
$\displaystyle=2R\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Bigg{|}X=t^{*}\right]+o(1)=O(R)$
(51)

where the second to last equality comes from Assumption 2.2. The envelope
function is uniformly square-integrable for $R$ near 0, and therefore, the
classes $\mathcal{G}_{R}$ are uniformly manageable.

  1. 4.

Define $h(t)=Pm(Z,t)$ and consider derivatives:

$\displaystyle h(t)=$
$\displaystyle\mathbb{E}_{P}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\})\right]=$
(52)
$\displaystyle\mathbb{E}_{P}\left[\left(Y_{1}-Y_{0}\right)(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\})\right]$
(53) $\displaystyle h^{\prime}(t)=$ $\displaystyle-
f_{x}(t)\mathbb{E}_{P}\left[Y_{1}-Y_{0}\Big{|}X=t\right]$ (54) $\displaystyle
h^{\prime\prime}(t)=$
$\displaystyle-f^{\prime}_{x}(t)\mathbb{E}_{P}\left[Y_{1}-Y_{0}\Big{|}X=t\right]-f_{x}(t)\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t\right]}{\partial
X}\right).$ (55)

Assumption 2.3 with $s=2$ guarantees the existence of $h^{\prime}$ and
$h^{\prime\prime}$. Since
$\mathbb{E}_{P}\left[Y_{1}-Y_{0}\Big{|}X=t^{*}\right]=0$, $H$ is given by

$\displaystyle
H=-h^{\prime\prime}(t^{*})=f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$ (56)

  2. 5.

This condition is divided into two parts. First, I prove the existence of
$K(s,r)=\lim_{\alpha\rightarrow\infty}\alpha
Pm\left(\cdot,t^{*}+r/\alpha\right)m\left(\cdot,t^{*}+s/\alpha\right)$ for
each $s,r$ in $\mathbb{R}$. Covariance $K$ is:

$\displaystyle Pm$
$\displaystyle\left(\cdot,t^{*}+r/\alpha\right)m\left(\cdot,t^{*}+s/\alpha\right)=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\right.$
(57)
$\displaystyle(\mathbf{1}\\{X>t^{*}+r/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})(\mathbf{1}\\{X>t^{*}+s/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})\Big{]}.$
(58)

If $rs<0$, covariance and $K(s,r)$ are 0. If $rs>0$, and suppose $r>0$:

$\displaystyle
Pm\left(\cdot,t^{*}+r/\alpha\right)m\left(\cdot,t^{*}+s/\alpha\right)=$ (59)
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X\in(t^{*},t^{*}+\min\\{r,s\\}/\alpha)\right]$
(60)

and hence

$\displaystyle K(s,r)$
$\displaystyle=\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X\in(t^{*},t^{*}+\min\\{r,s\\}/\alpha)\right]$
(61)
$\displaystyle=\min\\{r,s\\}f_{x}(t^{*})\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X=t^{*}\right].$
(62)

where the equality is due to continuity of $f_{x}$ (Assumption 2.3 with
$s=2$). Boundedness of the quantity follows from Assumptions 2.2 and 2.3.

Now, I will prove that $\lim_{\alpha\rightarrow\infty}\alpha
Pm\left(\cdot,t^{*}+r/\alpha\right)^{2}\mathbf{1}\left\\{\left|m\left(\cdot,t^{*}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}=0$
for each $\varepsilon>0$ and $r$ in $\mathbb{R}$. I have:

$\displaystyle\alpha
Pm\left(\cdot,t^{*}+r/\alpha\right)^{2}\mathbf{1}\left\\{\left|m\left(\cdot,t^{*}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}=$
(63)
$\displaystyle\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\right.$
(64)
$\displaystyle(\mathbf{1}\\{X>t^{*}+r/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})^{2}\mathbf{1}\left\\{\left|m\left(\cdot,t^{*}+r/\alpha\right)\right|>\varepsilon\alpha\right\\}\Big{]}=$
(65)
$\displaystyle\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\right.$
(66)
$\displaystyle(\mathbf{1}\\{X>t^{*}+r/\alpha\\}-\mathbf{1}\\{X>t^{*}\\})^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\varepsilon\alpha\right\\}\Big{]}\leq$
(67)
$\displaystyle\alpha\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\varepsilon\alpha\right\\}\right].$
(68)

Some algebra gives

$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\epsilon\right\\}\right]=$
(69)
$\displaystyle\mathbb{E}\left[\left(\frac{DY_{1}^{2}}{p(\textbf{X})^{2}}+\frac{(1-D)Y_{0}^{2}}{(1-p(\textbf{X}))^{2}}\right)\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>\epsilon\right\\}\right]=$
(70)
$\displaystyle\mathbb{E}\left[\frac{Y_{1}^{2}}{p(\textbf{X})}\mathbf{1}\left\\{\left|\frac{Y_{1}}{p(\textbf{X})}\right|>\epsilon\right\\}\right]+\mathbb{E}\left[\frac{Y_{0}^{2}}{(1-p(\textbf{X}))}\mathbf{1}\left\\{\left|\frac{Y_{0}}{(1-p(\textbf{X}))}\right|>\epsilon\right\\}\right]=$
(71)
$\displaystyle\frac{1}{p(\textbf{X})}\mathbb{E}\left[Y_{1}^{2}\mathbf{1}\left\\{\left|Y_{1}\right|>\epsilon_{1}\right\\}\right]+\frac{1}{(1-p(\textbf{X}))}\mathbb{E}\left[Y_{0}^{2}\mathbf{1}\left\\{\left|Y_{0}\right|>\epsilon_{0}\right\\}\right]$
(72)

and hence the condition is satisfied if

$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[Y_{1}^{2}\mathbf{1}\left\\{\left|Y_{1}\right|>\epsilon\alpha\right\\}\right]=0$
(73)
$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[Y_{0}^{2}\mathbf{1}\left\\{\left|Y_{0}\right|>\epsilon\alpha\right\\}\right]=0.$
(74)

Consider the limit for $Y_{1}$:

$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\mathbb{E}\left[Y_{1}^{2}\mathbf{1}\left\\{\left|Y_{1}\right|>\epsilon\alpha\right\\}\right]=\lim_{\alpha\rightarrow\infty}\frac{\int_{\epsilon\alpha}Y_{1}^{2}\varphi_{1}(y_{1})dy_{1}}{\alpha^{-1}}=\lim_{\alpha\rightarrow\infty}\frac{\epsilon^{3}\alpha^{2}\varphi_{1}(\epsilon\alpha)}{\alpha^{-2}}=$
(75)
$\displaystyle\lim_{\alpha\rightarrow\infty}\epsilon^{3}\alpha^{4}\varphi_{1}(\epsilon\alpha)=\lim_{\alpha\rightarrow\infty}\epsilon^{3}\alpha^{4}|\epsilon\alpha|^{-(4+\delta)}o(1)=0$
(76)

where the second to last equality follows from Assumption 3.

  3. 6.

I showed that $PG_{R}^{2}=O(R)$ as $R\rightarrow 0$ in equation (49). I need
to prove that for each $\varepsilon>0$ there is a constant C such that
$PG_{R}^{2}\mathbf{1}\\{G_{R}>C\\}\leq\varepsilon R$ for $R$ near 0. For any
$\varepsilon>0$ and $C>0$:

$\displaystyle PG_{R}^{2}$
$\displaystyle\mathbf{1}\\{G_{R}>C\\}=\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\\{\left|X-t^{*}\right|<R\\}\mathbf{1}\\{G_{R}>C\\}\right]\leq$
(77)
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\\{\left|X-t^{*}\right|<R\\}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>C\right\\}\right]\leq$
(78)
$\displaystyle\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\mathbf{1}\left\\{\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|>C\right\\}\right]\rightarrow
0$ (79)

where the last limit is taken for $C\rightarrow\infty$, and follows from
Assumption 3.

  4. 7.

I need to show that
$P\left|m\left(\cdot,t_{1}\right)-m\left(\cdot,t_{2}\right)\right|=O\left(\left|t_{1}-t_{2}\right|\right)$
near $t^{*}$. Consider $t_{2}>t_{1}$:

$\displaystyle
P\left|m\left(\cdot,t_{1}\right)-m\left(\cdot,t_{2}\right)\right|=\mathbb{E}\left[\left|\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)(\mathbf{1}\\{X>t_{1}\\}-\mathbf{1}\\{X>t_{2}\\})\right|\right]\leq$
(80) $\displaystyle
M_{x}\mathbb{E}\left[\left|\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\right|\Big{|}X\in(t_{1},t_{2})\right]\leq
M_{x}M_{y}|t_{2}-t_{1}|$ (81)

where $M_{x}=\max_{x\in(t_{1},t_{2})}f_{x}(x)$ and
$M_{y}=\max_{x\in(t_{1},t_{2})}\mathbb{E}\left[\left|\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\right|\Big{|}X\right]$.
$M_{x}<\infty$ and $M_{y}<\infty$ because of Assumption 2.3.

Assumptions 1-7 of Theorem Theorem Kim and Pollard by kim1990cube are hence
satisfied. It follows that, for $n\rightarrow\infty$,

$\displaystyle
n^{1/3}\left(\hat{t}^{e}_{n}-t^{*}\right)\rightarrow^{d}\mathop{\rm
arg~{}max}\limits_{r}Q(r)$ (82)

where $Q(r)=Q_{1}(r)+Q_{0}(r)$, and $Q_{1}$ is a non degenerate zero-mean
Gaussian process with covariance $K$, while $Q_{0}(r)$ is non-random and
$Q_{0}(r)=-\frac{1}{2}r^{2}H$.

Limiting distribution $\mathop{\rm arg~{}max}\limits_{r}Q(r)$ is of
chernoff1964estimation type. It can be shown (banerjee2001likelihood) that

$\displaystyle\mathop{\rm
arg~{}max}\limits_{r}Q(r)=^{d}(2\sqrt{K}/H)^{2/3}\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}$ (83)

where $B(r)$ is the two-sided Brownian motion process, $K$ is:

$\displaystyle K=$ $\displaystyle
f_{x}(t^{*})\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}\Big{|}X=t^{*}\right]$
(84) $\displaystyle=$ $\displaystyle
f_{x}(t^{*})\left(\frac{1}{p(\textbf{X})}\mathbb{E}[Y_{1}^{2}|X=t^{*}]+\frac{1}{1-p(\textbf{X})}\mathbb{E}[Y_{0}^{2}|X=t^{*}]\right)$
(85)

and $H$ is:

$\displaystyle
H=f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right).$ (86)

This completes the proof of the theorem. ∎

### Corollary 2.1

######  Corollary 2.1.

The asymptotic distribution of regret $\mathcal{R}(\hat{t}^{e}_{n})$ is:

$\displaystyle
n^{\frac{2}{3}}\mathcal{R}(\hat{t}^{e}_{n})\rightarrow^{d}\left(\frac{2K^{2}}{H}\right)^{\frac{1}{3}}\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}.$

The expected value of the asymptotic distribution is
$K^{\frac{2}{3}}H^{-\frac{1}{3}}C^{e}$, where

$C^{e}=\sqrt[3]{2}\mathbb{E}\left[\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}\right]$

is a constant not dependent on $P$.

###### Proof.

Result in equation (6) for $\hat{t}^{e}_{n}$ implies

$\displaystyle
n^{\frac{2}{3}}\mathcal{R}(\hat{t}^{e}_{n})=\frac{1}{2}W^{\prime\prime}(\tilde{t})\left(n^{\frac{1}{3}}\left(\hat{t}^{e}_{n}-t^{*}\right)\right)^{2},$

where $|\tilde{t}-t^{*}|\leq|\hat{t}_{n}-t^{*}|$. By continuous mapping
theorem

$\displaystyle
W^{\prime\prime}(\tilde{t})\rightarrow^{p}W^{\prime\prime}(t^{*})=H$

and hence by Slutsky’s theorem

$\displaystyle
n^{\frac{2}{3}}\left(W(\hat{t}_{n})-W(t^{*})\right)\rightarrow^{d}\left(\frac{2K^{2}}{H}\right)^{\frac{1}{3}}\left(\mathop{\rm
arg~{}max}\limits_{r}B(r)-r^{2}\right)^{2}.$

∎

### Theorem 3

######  Theorem 3.

Consider the SWM policy $\hat{t}^{s}_{n}$ defined in equation (11) and the
optimal policy $t^{*}$ defined in equation (4). Under Assumptions 1, 2 with
$s=0$ and 4,

$\displaystyle\hat{t}^{s}_{n}\rightarrow^{a.s.}t^{*}$

i.e. $\hat{t}^{s}_{n}$ is a consistent estimator for $t^{*}$.

###### Proof.

To prove the result, I show that conditions for Theorem 4.1.1 in
amemiya1985advanced hold, and hence $\hat{t}^{s}_{n}$ is consistent for
$t^{*}$.

First, define function $m^{s}(Z,t)$:

$\displaystyle
m^{s}\left(Z,t\right)=\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\left(k\left(\frac{X-t}{\sigma_{n}}\right)-k\left(\frac{X-t^{*}}{\sigma_{n}}\right)\right)$

and recall definitions of $m(Z,t)$, $P_{n}$, and $P$ introduced in the proof
of Theorem 2:

$\displaystyle m\left(Z,t\right)$
$\displaystyle=\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\left(\mathbf{1}\\{X>t\\}-\mathbf{1}\\{X>t^{*}\\}\right)$
(87) $\displaystyle P_{n}m\left(\cdot,t\right)$
$\displaystyle=\sum_{i=1}^{n}m\left(Z_{i},t\right)$ (88) $\displaystyle
Pm\left(Z,t\right)$ $\displaystyle=\mathbb{E}_{P}[m\left(Z,t\right)].$ (89)

In this notation, $\hat{t}^{s}_{n}=\mathop{\rm
arg~{}max}\limits_{t}P_{n}m^{s}(.,t)$ and $t^{*}=\mathop{\rm
arg~{}max}\limits_{t}Pm(Z,t)$. I can now show that conditions $A$, $B$, and
$C$ for Theorem 4.1.1 in amemiya1985advanced hold:

  * A)

Parameter space $\mathcal{T}$ is compact by Assumption 2.1.

  * B)

Function $P_{n}m^{s}\left(Z_{i},t\right)$ is continuous in $t$ for all $Z$ and
is a measurable function of $Z$ for all $t\in\mathcal{T}$, as $k(\cdot)$ is
continuous by Assumption 4.

  * C1)

I need to prove that $P_{n}m^{s}\left(Z_{i},t\right)$ converges a.s. to
$Pm(Z,t)$ uniformly in $t\in\mathcal{T}$ as $n\rightarrow\infty$, i.e.
$\sup_{t}\left|P_{n}m^{s}(\cdot,t)-Pm(Z,t)\right|\rightarrow^{a.s.}0$. Note
that:

$\displaystyle\sup_{t}\left|P_{n}m^{s}(\cdot,t)-Pm(Z,t)\right|\leq$ (90)
$\displaystyle\sup_{t}\left|P_{n}m^{s}(\cdot,t)-Pm^{s}(Z,t)\right|+\sup_{t}\left|Pm^{s}(\cdot,t)-Pm(Z,t)\right|.$
(91)

I need to show that the two addends on the right-hand side converge to zero.

To show uniform convergence of $P_{n}m^{s}(\cdot,t)$ to $Pm^{s}(Z,t)$, I
consider sufficient conditions provided by Theorem 4.2.1 in
amemiya1985advanced. $m^{s}(Z,t)$ is continuous in $t\in\mathcal{T}$ with
$\mathcal{T}$ compact, and measurable in $Z$. I only need to show that
$\mathbb{E}[\sup_{t\in\mathcal{T}}|m^{s}(Z,t)|]<\infty$. By Assumption 4,
$k(\cdot)$ is a bounded function, i.e. it exists an $M$ such that $|k(x)|<M$
for all $x$. Hence $\mathbb{E}[\sup_{t\in\mathcal{T}}|m^{s}(Z,t)|]\leq
M\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|\right]$,
and
$\mathbb{E}\left[\left|\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right|\right]<\infty$
by Assumption 2.2.

To show uniform convergence of $Pm^{s}(\cdot,t)$ to $Pm(Z,t)$, note that
$t\in\mathcal{T}$, where $\mathcal{T}$ is bounded, and hence the result holds
for $\sigma_{n}\rightarrow 0$.

  * C2)

By Assumption 2.1, $t^{*}$ is the unique global maximum of $Pm(Z,t)$.

Assumptions $A$, $B$, $C$ of Theorem 4.1.1 in amemiya1985advanced are
satisfied, and hence $\hat{t}^{s}_{n}\rightarrow^{a.s.}t^{*}$. ∎

### Lemmas

Proof of Theorem 4 requires some intermediate lemmas, stated and proved below.
Arguments follows the ideas in horowitz1992smoothed, but are adapted to my
context. I report the entire proof for completeness, even when it overlaps
with the original in horowitz1992smoothed.

To make the notation simpler, define:

$\displaystyle\hat{S}_{n}(t,\sigma_{n})=\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right]$

and note that $\hat{t}^{s}_{n}=\mathop{\rm
arg~{}max}\limits_{t}\hat{S}_{n}(t,\sigma_{n})$. Then define:

$\displaystyle\hat{S}_{n}^{1}(t,\sigma_{n})=\frac{\partial\hat{S}_{n}(t,\sigma_{n})}{\partial
t}=-\frac{1}{\sigma_{n}}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right]$
$\displaystyle\hat{S}_{n}^{2}(t,\sigma_{n})=\frac{\partial\hat{S}_{n}(t,\sigma_{n})}{\partial^{2}t}=\frac{1}{\sigma_{n}^{2}}\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime\prime}\left(\frac{X_{i}-t}{\sigma_{n}}\right)\right].$

Indicate with $\varphi_{y,x}(y,x)$ the joint distribution of $Y_{1}$, $Y_{0}$,
and $X$, and with $\varphi_{y|x}(y|x)$ the conditional distribution, where
$\varphi_{y,x}(y,x)=\varphi_{y|x}(y|x)f_{x}(x)$.

#### Lemma 1

######  Lemma 1.

Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, and 5 hold. Then

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=A$
(92)
$\displaystyle\lim_{n\rightarrow\infty}\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\alpha_{2}K.$
(93)

###### Proof.

First, I will prove that
$\lim_{n\rightarrow\infty}\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=A$:

$\displaystyle\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]$
$\displaystyle=-\frac{\sigma_{n}^{-h}}{\sigma_{n}}\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\right]=$
(94)
$\displaystyle=-\frac{\sigma_{n}^{-h}}{\sigma_{n}}\mathbb{E}\left[\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\right]=$
(95)
$\displaystyle=-\sigma_{n}^{-h}\int_{x}\int_{y}\left(Y_{1}-Y_{0}\right)\frac{1}{\sigma_{n}}k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\varphi_{y,x}(y,x)dydx=$
(96)
$\displaystyle=-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\varphi_{y,x}(y,t^{*}+\zeta\sigma_{n})dyd\zeta$
(97)

where in the last line I made the substitution
$\zeta=\frac{X_{i}-t^{*}}{\sigma_{n}}$. Consider the Taylor expansion of
$\varphi$ around $\varphi(y,t^{*})$:

$\displaystyle\varphi(y,t^{*}+\zeta\sigma_{n})$
$\displaystyle=\varphi(y,t^{*})+\zeta\sigma_{n}\varphi^{1}_{2}(y,t^{*})+\frac{1}{2}(\zeta\sigma_{n})^{2}\varphi^{2}_{2}(y,t^{*})+\dots$
(98)
$\displaystyle=\varphi(y,t^{*})+\left(\sum_{i=1}^{h-1}\frac{1}{i!}\varphi^{i}_{2}(y,x=t^{*})\zeta^{i}\sigma_{n}^{i}\right)+\frac{1}{h!}\varphi^{h}_{2}(y,\tilde{t})\zeta^{h}\sigma_{n}^{h}$
(99)

with $|\tilde{t}-t^{*}|\leq|t^{*}+\zeta\sigma_{n}-t^{*}|$. Existence of
$\varphi^{m}_{2}(y,x)$, the $m$-derivatives of $\varphi(y,x)$ with respect to
its second argument, is guaranteed by Assumption 2.3 with $s=h+1$.

Write
$\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]$ as
$I_{n1}+I_{n2}+I_{n3}$, where:

$\displaystyle I_{n1}=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\varphi(y,t^{*})dyd\zeta$
(100) $\displaystyle=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}k^{\prime}\left(\zeta\right)d\zeta\underbrace{\int_{y}\left(Y_{1}-Y_{0}\right)dy}_{=\mathbb{E}[Y_{1}-Y_{0}|X=t^{*}]=0}=0$
(101) $\displaystyle I_{n2}=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\left(\sum_{i=1}^{h-1}\frac{1}{i!}\varphi^{i}_{2}(y,x=t^{*})\zeta^{i}\sigma_{n}^{i}\right)dyd\zeta$
(102) $\displaystyle=$
$\displaystyle\sigma_{n}^{-h}\int_{y}\left(Y_{1}-Y_{0}\right)\left(\sum_{i=1}^{h-1}\frac{1}{i!}\varphi^{i}_{2}(y,x=t^{*})\sigma_{n}^{i}\underbrace{\int_{\zeta}k^{\prime}\left(\zeta\right)\zeta^{i}d\zeta}_{=0}\right)dy=0.$
(103)

Result on $I_{n1}$ follows from definition of $t^{*}$, while Assumption 5.2
guarantees result on $I_{n2}$. Finally, consider $I_{n3}$:

$\displaystyle I_{n3}=$
$\displaystyle-\sigma_{n}^{-h}\int_{\zeta}\int_{y}\left(Y_{1}-Y_{0}\right)k^{\prime}\left(\zeta\right)\frac{1}{h!}\varphi^{h}_{2}(y,\tilde{t})\zeta^{h}\sigma_{n}^{h}dyd\zeta$
(104) $\displaystyle=$
$\displaystyle-\frac{1}{h!}\int_{\zeta}k^{\prime}\left(\zeta\right)\zeta^{h}d\zeta\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{2}(y,\tilde{t})dy$
(105)

and conclude that:

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=-\frac{1}{h!}\int_{\zeta}\zeta^{h}k^{\prime}\left(\zeta\right)d\zeta\int_{y}\left(Y_{1}-Y_{0}\right)\varphi^{h}_{2}(y,t^{*})dy=A.$
(106)

Now I will prove that
$\lim_{n\rightarrow\infty}\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\alpha_{2}K$.
Note that:

$\displaystyle\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=$
$\displaystyle\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{-1/2}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}\right)\right]\right]=$
(107) $\displaystyle=$
$\displaystyle\sigma_{n}\operatorname{\mathrm{Var}}\left[\frac{1}{\sigma_{n}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)\right]=$
(108) $\displaystyle=$
$\displaystyle\sigma_{n}\mathbb{E}\left[\frac{1}{\sigma_{n}^{2}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)^{2}\right]-$
(109)
$\displaystyle\sigma_{n}\mathbb{E}\left[\frac{1}{\sigma_{n}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)\right]^{2}.$
(110)

Second term in the last expression goes to 0 as $n\rightarrow\infty$. For the
first term, observe that:

$\displaystyle\sigma_{n}\mathbb{E}\left[\frac{1}{\sigma_{n}}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)^{2}\right]=$
(111)
$\displaystyle\int_{x}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}\right)^{2}\frac{1}{\sigma_{n}}\varphi_{y,x}(y,x)dydx=$
(112)
$\displaystyle\int_{\zeta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}k^{\prime}\left(\zeta\right)^{2}\varphi_{y,x}(y,t^{*}+\zeta\sigma_{n})dyd\zeta$
(113)

where in the last line I made the substitution
$\zeta=\frac{X_{i}-t^{*}}{\sigma_{n}}$. Conclude that

$\displaystyle\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=$
$\displaystyle\int_{\zeta}k^{\prime}\left(\zeta\right)^{2}d\zeta
f_{x}(t^{*})\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)^{2}|X=t^{*}\right]$
(114) $\displaystyle=$ $\displaystyle\alpha_{2}K.$ (115)

Note that $\alpha_{2}K$ is bounded by Assumptions 2.2, 2.3, and 5.2. ∎

#### Lemma 2

######  Lemma 2.

Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, and 5 hold. If
$n\sigma_{n}^{2h+1}\rightarrow\infty$,
$\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})$ converges in probability to
A. If $n\sigma_{n}^{2h+1}$ has a finite limit $\lambda$,
$(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})$ converges in
distribution to $\mathcal{N}(\lambda^{1/2}A,\alpha_{2}K)$.

###### Proof.

Note that

$\operatorname{\mathrm{Var}}\left[(\sigma_{n})^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\underbrace{(n\sigma_{n}^{2h+1})^{-1}}_{\rightarrow
0}\underbrace{\operatorname{\mathrm{Var}}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]}_{\rightarrow\alpha_{2}K}.$

So the first result follows from lemma 1 and Chebyshev’s inequality.

For the second result, first note that under the stated assumptions and from
lemma 1,

$\mathbb{E}\left[(n\sigma_{n})^{1/2}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]=\underbrace{(n\sigma_{n}^{2h+1})^{1/2}}_{\rightarrow\lambda^{1/2}}\underbrace{\mathbb{E}\left[\sigma_{n}^{-h}\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]}_{A}$

and so the result follows if I show that

$U_{n}=(n\sigma_{n})^{1/2}\left(\hat{S}_{n}^{1}(t^{*},\sigma_{n})-\mathbb{E}\left[\hat{S}_{n}^{1}(t^{*},\sigma_{n})\right]\right)\rightarrow^{d}\mathcal{N}(0,\alpha_{2}K).$

Note that

$\displaystyle U_{n}=$
$\displaystyle(n\sigma_{n})^{1/2}\frac{1}{n}\sum_{i=1}^{n}\left[\underbrace{\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t}{\sigma_{n}}\right)\frac{1}{\sigma_{n}}}_{=B}-\mathbb{E}[B]\right]=$
(116) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])$
(117)

and hence $U_{n}$ has characteristic function $\psi(\tau)^{n}$, where

$\psi(\tau)=\mathbb{E}\left[\exp\left(i\tau\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\right)\right]$

and

$\displaystyle\psi^{{}^{\prime}}(\tau)=\mathbb{E}\left[i\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\exp\left(i\tau\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\right)\right]$
(118)
$\displaystyle\psi^{{}^{\prime\prime}}(\tau)=\mathbb{E}\left[-\frac{\sigma_{n}}{n}(B-\mathbb{E}[B])\exp\left(i\tau\left(\frac{\sigma_{n}}{n}\right)^{1/2}(B-\mathbb{E}[B])\right)\right].$
(119)

Note that $\psi^{{}^{\prime}}(0)=0$ and
$\psi^{{}^{\prime\prime}}(0)=-\frac{\sigma_{n}}{n}\operatorname{\mathrm{Var}}(B)=-\frac{1}{n}(\alpha_{2}K+o(1))$,
since lemma 1 proved that
$\lim_{n\rightarrow\infty}\sigma_{n}\operatorname{\mathrm{Var}}(B)=\alpha_{2}K$.

A Taylor series expansion of $\psi(\tau)$ about $\tau=0$ yields:

$\psi(\tau)=\underbrace{\psi(0)}_{=1}+\underbrace{\psi^{{}^{\prime}}(0)}_{=0}\tau+\frac{1}{2}\underbrace{\psi^{{}^{\prime\prime}}(0)}_{=-\frac{\alpha_{2}K}{n}}\tau^{2}+o\left(\frac{\tau^{2}}{n}\right)=1-\frac{1}{2n}\alpha_{2}K\tau^{2}+o\left(\frac{\tau^{2}}{n}\right)$

and hence the characteristic function of $U_{n}$ has limit:

$\displaystyle\lim_{n\rightarrow\infty}\left[1-\frac{1}{2n}\alpha_{2}K\tau^{2}+o\left(\frac{\tau^{2}}{n}\right)\right]^{n}=\exp\left(-\alpha_{2}K\frac{\tau^{2}}{2}\right).$
(120)

Since $\exp\left(-\alpha_{2}K\frac{\tau^{2}}{2}\right)$ is the characteristic
function of $\mathcal{N}(0,\alpha_{2}K)$, the second result of the lemma
holds. ∎

#### Lemma 3

######  Lemma 3.

Let Assumptions 1, 2 with $s=h+1$ for some $h\geq 2$, and 5 hold. Let $\eta>0$
be such that $\varphi_{y,x}(y,x)$ has second derivative uniformly bounded for
almost every $X$ if $|X-t^{*}|<\eta$. For $\theta\in\mathbb{R}$, define
$\hat{S}^{\theta}_{n}(\theta)$ by

$\displaystyle\hat{S}^{\theta}_{n}(\theta)=-(n\sigma_{n}^{2})^{-1}\sum_{i=1}^{n}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)\right].$
(121)

Define the sets $\Theta_{n}(n=1,2,\dots)$ by
$\Theta_{n}=\\{\theta:\theta\in\mathbb{R},\sigma_{n}|\theta|\leq\frac{\eta}{2}\\}$.
Then

$\displaystyle\operatorname{\mathrm{plim}}_{n\rightarrow\infty}\sup_{\theta\in\Theta_{n}}|\hat{S}^{\theta}_{n}(\theta)-\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]|=0.$
(122)

In addition, there are finite numbers $\alpha_{1}$ and $\alpha_{2}$ such that
for all $\theta\in\Theta_{n}$

$|\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]-H\theta|\leq
o(1)+\alpha_{1}\sigma_{n}|\theta|+\alpha_{2}\sigma_{n}\theta^{2}$

uniformly over $\theta\in\Theta_{n}$.

###### Proof.

To prove the first result, first define

$\displaystyle-g_{i}(\theta)=$
$\displaystyle\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)-$
(123)
$\displaystyle\mathbb{E}\left[\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)\right].$
(124)

It is necessary to prove that for any $\varepsilon>0$

$\lim_{n\rightarrow\infty}\operatorname{Pr}\left[\sup_{\theta\in\Theta_{n}}\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{n=1}^{N}g_{i}(\theta)\right|>\varepsilon\right]=0.$

Given any $\delta>0$, divide each set $\Theta_{n}$ into nonoverlapping subsets
$\Theta_{nj}(j=1,2,\ldots)$ such that the distance between any two points in
the same subset does not exceed $\delta\sigma_{n}^{2}$ and the number
$\Gamma_{n}$ of subsets does not exceed $C\sigma_{n}^{-3(q-1)}$ for some
$C>0$. Let $\left\\{\theta_{Ni}\right\\}$ be a set of vectors such that
$\theta_{nj}\in\Theta_{nj}$. Then

$\displaystyle\operatorname{Pr}\left[\sup_{\theta\in\Theta_{n}}\left(n\sigma_{n}^{2}\right)^{-1}\right.$
$\displaystyle\left.\left|\sum_{n=1}^{n}g_{i}(\theta)\right|>\varepsilon\right]=$
(125) $\displaystyle=$
$\displaystyle\operatorname{Pr}\left[\bigcup_{j=1}^{\Gamma_{n}}\sup_{\theta\in\Theta_{nj}}\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}(\theta)\right|>\varepsilon\right]$
(126) $\displaystyle\leqslant$
$\displaystyle\sum_{j=1}^{\Gamma_{n}}\operatorname{Pr}\left[\sup_{\theta\in\Theta_{nj}}\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}(\theta)\right|>\varepsilon\right]$
(127) $\displaystyle\leqslant$
$\displaystyle\underbrace{\sum_{j=1}^{\Gamma_{n}}\operatorname{Pr}\left[\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}\left(\theta_{nj}\right)\right|>\varepsilon/2\right]}_{=B_{1}}$
(128)
$\displaystyle+\underbrace{\sum_{j=1}^{\Gamma_{n}}\operatorname{Pr}\left[\left(n\sigma_{n}^{2}\right)^{-1}\sum_{i=1}^{n}\sup_{\theta\in\Theta_{nj}}\left|g_{i}(\theta)-g_{i}\left(\theta_{nj}\right)\right|>\varepsilon/2\right]}_{=B_{2}},$
(129)

where the last two lines follow from the triangle inequality. By Hoeffding’s
inequality, there are finite numbers $c_{1}>0$ and $c_{2}>0$ such that

$\operatorname{Pr}\left[\left(n\sigma_{n}^{2}\right)^{-1}\left|\sum_{i=1}^{n}g_{i}\left(\theta_{nj}\right)\right|>\varepsilon/2\right]\leqslant
c_{1}\exp\left(-c_{2}n\sigma_{n}^{4}\right).$

Therefore, $B_{1}$ is bounded by
$Cc_{1}\sigma_{n}^{-3(q-1)}\exp\left(-c_{2}n\sigma_{n}^{4}\right)$, which
converges to 0 as $n\rightarrow\infty$ by Assumption 5.1. In addition, by
Assumption 4 there is a finite $c_{3}>0$ such that if $\theta\in\Theta_{nj}$,

$\displaystyle\left|-\left(\frac{D_{i}Y_{i}}{p(\textbf{X}_{i})}-\frac{(1-D_{i})Y_{i}}{(1-p(\textbf{X}_{i}))}\right)\left[k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta\right)-k^{\prime}\left(\frac{X_{i}-t^{*}}{\sigma_{n}}+\theta_{nj}\right)\right]\right|$
(130) $\displaystyle\leqslant c_{3}\left|\theta-\theta_{nj}\right|\leqslant
c_{3}\delta\sigma_{n}^{2}.$ (131)

So

$\left(n\sigma_{n}^{2}\right)^{-1}\sum_{i=1}^{n}\sup_{\theta\in\Theta_{nj}}\left|g_{i}(\theta)-g_{i}\left(\theta_{nj}\right)\right|\leqslant
2c_{3}\delta.$

Choose $\delta<\varepsilon/4c_{3}$. Then $B_{2}$ is 0. This establishes
$\operatorname{\mathrm{plim}}_{n\rightarrow\infty}\sup_{\theta\in\Theta_{n}}|\hat{S}^{\theta}_{n}(\theta)-\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]|=0$.

To prove the second result, start noting that

$\displaystyle\mathbb{E}[\hat{S}^{\theta}_{n}(\theta)]=-\sigma_{n}^{-2}\mathbb{E}\left[\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\right]=I_{n1}+I_{n2}$
(132)

where

$\displaystyle
I_{n1}=-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,x)dydx$
(133) $\displaystyle
I_{n2}=-\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,x)dydx.$
(134)

First, consider $I_{n2}$ and observe that

$\displaystyle
I_{n2}=-\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\underbrace{\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\varphi(y|x)dy}_{=\mathbb{E}[Y_{1}-Y_{0}|X]}f_{x}(x)dx$
(135)

and since $\mathbb{E}[Y_{1}-Y_{0}|X]$ is bounded by Assumption 2.2,

$|I_{n2}|\leq\left|C\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)f_{x}(x)dx\right|.$

Define $\zeta=\frac{X-t^{*}}{\sigma_{n}}+\theta$. Since
$\sigma_{n}|\theta|\leq\frac{\eta}{2}$, when $|X-t^{*}|>\eta$

$\displaystyle|\zeta|=$
$\displaystyle\left|\frac{X-t^{*}}{\sigma_{n}}+\theta\right|=\pm\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\geq\pm\left(\frac{X-t^{*}}{\sigma_{n}}\right)-|\theta|=\left|\frac{X-t^{*}}{\sigma_{n}}\right|-|\theta|$
(136) $\displaystyle\geq$
$\displaystyle\left|\frac{X-t^{*}}{\sigma_{n}}\right|-\frac{\eta}{2\sigma_{n}}>\frac{\eta}{\sigma_{n}}-\frac{\eta}{2\sigma_{n}}=\frac{\eta}{2\sigma_{n}}.$
(137)

and so the event $|X-t^{*}|>\eta$ implies $|\zeta|>\frac{\eta}{2\sigma_{n}}$.
Then

$\displaystyle|I_{n2}|\leq$
$\displaystyle\left|C\sigma_{n}^{-2}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)f_{x}(x)dx\right|$
(138) $\displaystyle=$
$\displaystyle\left|C\sigma_{n}^{-1}\int_{|X-t^{*}|>\eta}k^{\prime}\left(\zeta\right)f_{x}(t^{*}-\theta\sigma_{n})d\zeta\right|$
(139) $\displaystyle\leq$
$\displaystyle\left|C\underbrace{f_{x}(t^{*}-\theta\sigma_{n})}_{\rightarrow
f_{x}(t^{*})}\underbrace{\sigma_{n}^{-1}\int_{|\zeta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta}_{\rightarrow
0}\right|.$ (140)

The fact that $f_{x}(t^{*}-\theta\sigma_{n})\rightarrow f_{x}(t^{*})$ bounded
by Assumption 2.3 with $s=h+1$ and
$\sigma_{n}^{-1}\int_{|\zeta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\rightarrow
0$ by Assumption 5.2 implies

$\operatorname{\mathrm{plim}}_{n\rightarrow\infty}\sup_{\theta\in\Theta_{n}}|I_{n2}|=0.$

Recall that $I_{n1}$ is defined as

$I_{n1}=-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,x)dydx.$

Consider a Taylor expansion of $\varphi(y,x)$ about $x=t^{*}$:

$\varphi(y,x)=\varphi(y,t^{*})+\varphi^{\prime}(y,t^{*})(x-t^{*})+\frac{1}{2}\varphi^{\prime\prime}(y,\tilde{t})(x-t^{*})^{2}$

with $|\tilde{t}-t^{*}|\leq|x-t^{*}|$. Write $I_{n1}$ as
$J_{n1}+J_{n2}+J_{n3}$ where

$\displaystyle J_{n1}=$
$\displaystyle-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi(y,t^{*})dydx$
(141) $\displaystyle=$
$\displaystyle-\sigma_{n}^{-2}\underbrace{\mathbb{E}[Y_{1}-Y_{0}|X=t^{*}]}_{=0}\int_{|X-t^{*}|\leq\eta}k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)f_{x}(t^{*})dx=0$
(142) $\displaystyle J_{n2}=$
$\displaystyle-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\varphi^{\prime}(y,t^{*})(x-t^{*})dydx$
(143) $\displaystyle J_{n3}=$
$\displaystyle-\sigma_{n}^{-2}\int_{|X-t^{*}|\leq\eta}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\frac{X-t^{*}}{\sigma_{n}}+\theta\right)\frac{1}{2}\varphi^{\prime\prime}(y,\tilde{t})(x-t^{*})^{2}dydx.$
(144)

Consider $J_{n2}$ and the substitution
$\zeta=\frac{X-t^{*}}{\sigma_{n}}+\theta$:

$\displaystyle J_{n2}=$
$\displaystyle-\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\zeta\right)(\zeta-\theta)\varphi^{\prime}(y,t^{*})dyd\zeta$
(145) $\displaystyle=$
$\displaystyle-\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\zeta\right)\zeta\varphi^{\prime}(y,t^{*})dyd\zeta+$
(146)
$\displaystyle\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)k^{\prime}\left(\zeta\right)\theta\varphi^{\prime}(y,t^{*})dyd\zeta$
(147) $\displaystyle=$
$\displaystyle-\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\underbrace{\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\varphi^{\prime}(y,t^{*})dy}_{=H}+$
(148)
$\displaystyle\theta\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\underbrace{\int_{y}\left(\frac{DY}{p(\textbf{X})}-\frac{(1-D)Y}{(1-p(\textbf{X}))}\right)\varphi^{\prime}(y,t^{*})dy}_{=H}$
(149)

where
$H=f_{x}(t^{*})\left(\frac{\partial\mathbb{E}_{P}\left[Y_{1}-Y_{0}|X=t^{*}\right]}{\partial
X}\right)$ is bounded by Assumption 2.3. Since $\int\zeta
k^{\prime}\left(\zeta\right)d\zeta=0$ and
$\sigma_{n}|\theta|\leq\frac{\eta}{2}$:

$\displaystyle\left|\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|=\left|\int_{|\zeta-\theta|>\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|\leq\left|\int_{|\zeta|>\eta/2\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|.$ (150)

By Assumption 5.2, $\left|\int_{|\zeta|>\eta/2\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta\right|$ converges to 0 uniformly over
$\theta\in\Theta_{n}$. It means that
$\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}\zeta
k^{\prime}\left(\zeta\right)d\zeta$ converges uniformly to 0.

Consider $\theta
H\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta$,
and note that, since $\int k^{\prime}\left(\zeta\right)d\zeta=1$,

$\displaystyle\left|\theta H-\theta
H\int_{|\zeta-\theta|\leq\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\right|=\left|\theta
H\int_{|\zeta-\theta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\right|\leq$
(151) $\displaystyle|\sigma_{n}\theta
H|\sigma_{n}^{-1}\int_{|\zeta-\theta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta\leq\frac{\eta}{2}\sigma_{n}^{-1}\int_{|\zeta-\theta|>\eta/\sigma_{n}}k^{\prime}\left(\zeta\right)d\zeta.$
(152)

The last term is bounded uniformly over $n$ and $\theta\in\Theta_{n}$ and
converges to 0 by Assumption 5.2. It means that

$\displaystyle\lim_{n\rightarrow\infty}\left|\sup_{\theta\in\Theta_{n}}J_{n2}-\theta
H\right|=0.$ (153)

Finally, consider $J_{n3}$:

$\displaystyle|J_{n3}|=$