Quantile Treatment Effects and Bootstrap Inference under Covariate-Adaptive Randomization
Yichong Zhang Xin Zheng

	 
	Magnetically tunable multi-band near-field radiative heat transfer
between two graphene sheets
     Ying Wu
    December 30, 2023
==============================================================================================


	In this paper, we study the estimation and inference of the quantile treatment effect under covariate-adaptive randomization. We propose two estimation methods: (1) the simple quantile regression and (2) the inverse propensity score weighted quantile regression. For the two estimators, we derive their asymptotic distributions uniformly over a compact set of quantile indexes, and show that, when the treatment assignment rule does not achieve strong balance, the inverse propensity score weighted estimator has a smaller asymptotic variance than the simple quantile regression estimator. For the inference of method (1), we show that the Wald test using a weighted bootstrap standard error under-rejects. But for method (2), its asymptotic size equals the nominal level. We also show that, for both methods, the asymptotic size of the Wald test using a covariate-adaptive bootstrap standard error equals the nominal level. We illustrate the finite sample performance of the new estimation and inference methods using both simulated and real datasets. 

	
	Keywords: Bootstrap inference, quantile treatment effectJEL codes: C14, C21






§ INTRODUCTION

The randomized control trial (RCT), as pointed out by <cit.>, is one of the five most common methods (along with instrumental variable regressions, matching estimations, differences-in-differences, and regression discontinuity designs) for causal inference. Researchers can use the RCT to estimate not only average treatment effects (ATEs) but also quantile treatment effects (QTEs), which capture the heterogeneity of the sign and magnitude of treatment effects, varying depending on their place in the overall distribution of outcomes. For example, <cit.> estimate the QTE of teacher performance pay program on student learning via the difference of empirical quantiles of test scores between treatment and control groups. <cit.> and <cit.> estimate the QTEs of audits on endline pollution and a group-lending microcredit program on informal borrowing, respectively, via linear quantile regressions (QRs). <cit.> estimate the QTE of microcredit on various household outcomes via a minimum distance method. <cit.> estimate the QTE of being informed on energy use via the inverse propensity score weighted (IPW) QR. Except <cit.>, the other four papers all use the bootstrap to construct confidence intervals for their QTE estimates. However, RCTs have also been routinely implemented with covariate-adaptive randomization. Individuals are first stratified based on some baseline covariates, and then, within each stratum, the treatment status is assigned (independent of covariates) to achieve some balance between the sizes of treatment and control groups; as examples, see <cit.> for a textbook treatment of the topic, and <cit.> and <cit.> for two excellent surveys on implementing RCTs in development economics. To achieve such balance, treatment status for different individuals usually exhibits a (negative) cross-sectional dependence. The standard inference procedures that rely on cross-sectional independence are usually conservative and lacking power. How do we consistently estimate QTEs under covariate-adaptive randomization? What are the asymptotic distributions for the QTE estimators, and how do we conduct proper bootstrap inference? These questions are as yet unaddressed.  

We propose two ways to estimate QTEs: (1) the simple quantile regression (SQR) and (2) the IPW QR. We establish the weak limits for both estimators uniformly over a compact set of quantile indexes and show that the IPW estimator has a smaller asymptotic variance than the SQR estimator when the treatment assignment rule does not achieve strong balance.[We will define “strong balance" in Section <ref>.] If strong balance is achieved, then the two estimators are asymptotically first-order equivalent. For inference, we show that the Wald test combined with weighted bootstrap based critical values can lead to under-rejection for method (1), but its asymptotic size equals the nominal level for method (2). We also study the covariate-adaptive bootstrap which respects the cross-sectional dependence when generating the bootstrap sample. The estimator based on the covariate-adaptive bootstrap sample can mimic that of the original sample in terms of the standard error. Thus, using proper covariate-adaptive bootstrap based critical values, the asymptotic size of the Wald test equals the nominal level for both estimators.  

As originally proposed by <cit.>, the QTE, for a fixed quantile index, corresponds to the horizontal difference between the marginal distributions of the potential outcomes for treatment and control groups. <cit.> studies the identification and estimation of QTE under unconfoundedness. Our estimators (1) and (2) directly follow those in <cit.> and <cit.>, respectively.

<cit.> first point out that, under covariate-adaptive randomization, the usual two-sample t-test for the ATE is conservative. They then propose a covariate-adaptive bootstrap which can produce the correct standard error. <cit.> extend the results to generalized linear models. However, both groups of researchers parametrize the (transformed) conditional mean equation by a specific linear model and focus on a specific randomization scheme (covariate-adaptive biased coin method). <cit.> derive the theoretical properties of ATE estimators based on general covariate-adaptive randomization under the linear model framework. <cit.> substantially generalize the framework to a fully nonparametric setting with a general class of randomization schemes. However, they mainly focus on the ATE and show that the standard two-sample t-test and the t-test based on the linear regression with strata fixed effects are conservative. They then obtain analytical estimators for the correct standard errors and study the validity of permutation tests. <cit.> study the IPW estimator for the ATE under adaptive randomization. However, they assume the treatment status is assigned completely independently across individuals. More recently, <cit.> study the estimation of ATE with multiple treatments and propose a fully saturated estimator. <cit.> study the estimation of ATE under an adaptive randomization procedure. 

Our paper complements the above papers in four aspects. First, we consider the estimation and inference of the QTE, which is a function of quantile index τ. We rely on the empirical processes theories developed by <cit.> and <cit.> to obtain uniformly weak convergence of our estimators over a compact set of τ. Based on the uniform convergence, we can construct not only point-wise but also uniform confidence bands. Second, we study the asymptotic properties of the IPW estimator under covariate-adaptive randomization. When the treatment assignment rule does not achieve strong balance, the IPW estimator is more efficient than the SQR estimator. Third, we investigate the weighted bootstrap approximation to the asymptotic distributions of the SQR and IPW estimators. We show that the weighted bootstrap ignores the (negative) cross-sectional dependence due to the covariate-adaptive randomization and over-estimates the asymptotic variance for the SQR estimator. However, the asymptotic variance for the IPW estimator does not rely on the randomization scheme implemented. Thus, the asymptotic size of the Wald test using the IPW estimator paired with the weighted bootstrap based critical values equals the nominal level. Fourth, we investigate the covariate-adaptive bootstrap approximation to the asymptotic distributions of the SQR and IPW estimators. We establish that, using either estimator paired with its corresponding covariate-adaptive bootstrap based critical values, the asymptotic size of the Wald test  equals the nominal level. <cit.> first propose the covariate-adaptive bootstrap and establish its validity for the ATE in a linear regression model under the null hypothesis that the treatment effect is not only zero but also homogeneous.[We say the average treatment effect is homogeneous if the conditional average treatment effect given covariates is the same as the unconditional one.] We modify the covariate-adaptive bootstrap and establish its validity for the QTE in the nonparametric setting proposed by <cit.>. In addition, our results do not rely on the homogeneity of the treatment effect. Compared with the analytical inference, the two bootstrap inferences for QTEs we study in this paper avoid estimating the infinite-dimensional nuisance parameters such as the densities of the potential outcomes, and thus, the choices of tuning parameters. In addition, unlike the permutation tests studied in <cit.>, the validity of bootstrap inferences does not require either strong balance condition or studentization. In particular, such studentization is cumbersome in the QTE context.  

As the asymptotic variance for the IPW estimator does not depend on the treatment assignment rule implemented in RCTs, this estimator (and equivalently, the fully saturated estimator for the ATE) is suitable for settings where the knowledge of the exact treatment assignment rule is not available. Such scenario occurs when researchers are using an experiment that was run in the past and the randomization procedure may not have been fully described. It also occurs in subsample analysis, where sub-groups are defined using variables that may have not been used to form the strata and the treatment assignment rule for each sub-group becomes unknown. We illustrate this fact in the subsample analysis of the empirical application in Section <ref>.     


The rest of the paper is organized as follows. In Section <ref>, we describe the model setup and notation. In Sections <ref> and <ref>, we discuss the asymptotic properties of estimators (1) and (2), respectively. In Sections <ref> and <ref>, we investigate the  weighted and covariate-adaptive bootstrap approximations to the asymptotic distributions of estimators (1) and (2), respectively. In Section <ref>, we examine the finite-sample performance of the estimation and inference methods. In Section <ref>, we  provide recommendations for practitioners. In Section <ref>, we apply the new methods to estimate and infer the average and quantile treatment effects of iron efficiency on educational attainment. In Section <ref>, we conclude. We provide proofs for all results in an appendix.  We study the strata fixed effects quantile regression estimator and provide additional simulation results in the second online supplement.  







§ SETUP AND NOTATION



First, denote the potential outcomes for treated and control groups as Y(1) and Y(0), respectively. The treatment status is denoted as A, where A=1 means treated and A=0 means untreated. The researcher can only observe {Y_i,Z_i,A_i}_i=1^n where Y_i = Y_i(1)A_i + Y_i(0)(1-A_i), and Z_i is a collection of baseline covariates. Strata are constructed from Z using a function S: (Z)↦𝒮, where 𝒮 is a finite set. For 1≤ i ≤ n, let S_i = S(Z_i) and p(s) = ℙ(S_i = s). Throughout the paper, we maintain the assumption that p(s) is fixed w.r.t. n and is positive for every s ∈𝒮.[We can also allow for the DGP to depend on n so that p_n(s) = ℙ_n(S_i = s) and p(s) = lim p_n(s). All the results in this paper still hold as long as n(s)→∞ a.s. Interested readers can refer to the previous version of this paper on arXiv for more detail.] We make the following assumption for the data generating process (DGP) and the treatment assignment rule.

		
	
		
  * {Y_i(1),Y_i(0),S_i}_i=1^n is i.i.d. 
		
  * {Y_i(1),Y_i(0)}^n_i=1{A_i}^n_i=1|{S_i}^n_i=1.
		
  * {{D_n(s)/√(n)}_s ∈𝒮|{S_i}^n_i=1} N(0,Σ_D) a.s., where 
		
    D_n(s) = ∑_i =1^n (A_i-π)1{S_i = s}  and  Σ_D = {p(s)γ(s):s ∈𝒮}

		with 0 ≤γ(s) ≤π(1-π).
		
  * D_n(s)/n(s) = o_p(1) for s ∈𝒮, where n(s) =∑_i =1^n1{S_i=s}. 
	


Several remarks are in order. First, Assumptions <ref>.1–<ref>.3 are exactly the same as <cit.>. We refer interested readers to <cit.> for more discussion of these assumptions. Second, note that in Assumption <ref>.3 the parameter π is the target proportion of treatment for each stratum and D_n(s) measures the imbalance. <cit.> study the more general case that π can take distinct values for different strata. Third, we follow the terminology in <cit.>, which follows that of <cit.> and <cit.>, saying a treatment assignment rule achieves strong balance if γ(s)=0. Fourth, we do not require that the treatment status is assigned independently. Instead, we only require Assumption <ref>.3 or Assumption <ref>.4, which condition is satisfied by several treatment assignment rules such as simple random sampling (SRS), biased-coin design (BCD), adaptive biased-coin design (WEI), and stratified block randomization (SBR). <cit.> provide an excellent summary of these four examples. For completeness, we briefly repeat their descriptions below. Note that both BCD and SBR assignment rules achieve strong balance. Last, as p(s)>0, Assumption <ref>.3 implies Assumption <ref>.4. 


[SRS]
	
	Let {A_i}_i=1^n be drawn independently across i and of {S_i}_i=1^n as Bernoulli random variables with success rate π, i.e., for k=1,⋯,n, 
	
    ℙ(A_k = 1|{S_i}_i=1^n, {A_j}_j=1^k-1) = ℙ(A_k = 1) = π.

	Then, Assumption <ref>.3 holds with γ(s) = π(1-π).




[WEI]
	
	The design is first proposed by <cit.>. Let n_k-1(S_k) = ∑_i=1^k-11{S_i = S_k}, D_k-1(s) = ∑_i=1^k-1(A_i - 1/2) 1{S_i = s}, and  
	
    ℙ(A_k = 1| {S_i}_i=1^k,{A_i}_i=1^k-1) = ϕ(D_k-1(S_k)/n_k-1(S_k)),

	where ϕ(·):[-1,1] ↦ [0,1] is a pre-specified non-increasing function satisfying ϕ(-x) = 1- ϕ(x). Here, D_0(S_1)/0 is understood to be zero. Then, <cit.> show that Assumption <ref>.3 holds with π = 1/2 and γ(s) = 1/4(1 - 4ϕ'(0))^-1.
 

[BCD]
	
	The treatment status is determined sequentially for 1 ≤ k ≤ n as
	
    ℙ(A_k = 1| {S_i}_i=1^k,{A_i}_i=1^k-1) = 1/2   if D_k-1(S_k) = 0 
    λ   if D_k-1(S_k) < 0 
    
    	1-λ   if D_k-1(S_k) > 0,

	where D_k-1(s) is defined as above and 1/2< λ≤ 1. Then, <cit.> show that Assumption <ref>.3 holds with π = 1/2 and γ(s) = 0. 	



[SBR]
	
	For each stratum, ⌊π n(s) ⌋ units are assigned to treatment and the rest is assigned to control. <cit.> then show that Assumption <ref>.3 holds with γ(s) = 0. 


Our parameter of interest is the τ-th QTE defined as 

    q(τ) = q_1(τ) - q_0(τ),

where τ∈ (0,1) is a quantile index and q_j(τ) is the τ-th quantile of random variable Y(j) for j = 0,1. For inference, although we mainly focus on the Wald test for the null hypothesis that q(τ) equals some particular value, our method can also be used to test hypotheses involving multiple or even a continuum of quantile indexes. The following regularity conditions are common in the literature of quantile estimations. 

	
	For j=0,1, denote f_j(·) and f_j(·|s) as the PDFs of Y_i(j) and Y_i(j)|S_i=s, respectively.
	
		
  * f_j(q_j(τ)) and f_j(q_j(τ)|s) are bounded and bounded away from zero uniformly over τ∈Υ and s ∈𝒮, where Υ is a compact subset of (0,1).
		
  * f_j(·) and f_j(·|s) are Lipschitz over {q_j(τ):τ∈Υ}.
	





§ ESTIMATION





 §.§ Simple Quantile Regression


In this section, we propose to estimate q(τ) by a QR of Y_i on A_i. Denote β(τ) = (β_0(τ),β_1(τ))^', β_0(τ) = q_0(τ), and β_1(τ) = q(τ). We estimate β(τ) by β̂(τ), where 

    β̂(τ) = _b = (b_0,b_1)^'∈^2∑_i=1^nρ_τ(Y_i - Ȧ_i'b),

Ȧ_i = (1,A_i)^', and ρ_τ(u) = u(τ - 1{u≤ 0}) is the standard check function. We refer to β̂_1(τ), the second element of β̂(τ), as our SQR estimator for the τ-th QTE. As A_i is a dummy variable, β̂_1(τ) is numerically the same as the difference between the τ-th empirical quantiles of Y in the treatment and control groups. 


	
	If Assumptions <ref>.1–<ref>.3 and <ref> hold, then, uniformly over τ∈Υ, 
	
    √(n)(β̂_1(τ) - q(τ)) ℬ_sqr(τ), as n→∞,

	where ℬ_sqr(·) is a Gaussian process with covariance kernel Σ_sqr(·,·). The expression for Σ_sqr(·,·) can be found in the Appendix. 

The asymptotic variance for √(n)(β̂_1(τ) - β_1(τ)) is ζ_Y^2(π,τ)  + ζ_A^2(π,τ) + ζ_S^2(τ), where

    ζ_Y^2(π,τ) = τ (1-τ) - 𝔼m_1^2(S,τ)/π f_1^2(q_1(τ)) + τ (1-τ) - 𝔼m_0^2(S,τ)/(1-π) f_0^2(q_0(τ)),


    ζ_A^2(π,τ) = 𝔼γ(S) (m_1(S,τ)/π f_1(q_1(τ)) + m_0(S,τ)/(1-π) f_0(q_0(τ)))^2,


    ζ_S^2(τ) = 𝔼(m_1(S,τ)/f_1(q_1(τ))-m_0(S,τ)/ f_0(q_0(τ)))^2,

and m_j(s,τ) = 𝔼(τ - 1{Y(j)≤ q_j(τ)}|S=s). Note that, if the treatment assignment rule achieves strong balance or the stratification is irrelevant[It means ℙ(Y(j) ≤ q_j(τ)|S=s) = τ for s ∈𝒮, j=0,1.] then ζ_A^2(π,τ) = 0. 









 §.§ Inverse Propensity Score weighted Quantile Regression


Denote π̂(s) = n_1(s)/n(s), n_1(s) = ∑_i=1^n A_i 1{S_i = s}, and n(s) = ∑_i=1^n 1{S_i = s}. Note π̂(S_i) is an estimator for the propensity score, i.e., π. In addition,  Assumption <ref>.2 implies that the unconfoundedness condition holds. Thus, following the lead of <cit.>, we can estimate q_j(τ) by the IPW QR. Let 

    q̂_1(τ) = _q 1/n∑_i=1^nA_i/π̂(S_i)ρ_τ(Y_i - q)   and  q̂_0(τ) = _q1/n∑_i=1^n 1-A_i/1-π̂(S_i)ρ_τ(Y_i - q).

We then estimate q(τ) by q̂(τ) = q̂_1(τ) - q̂_0(τ). 

	
	If Assumptions <ref>.1, <ref>.2, <ref>.4 and <ref> hold, then, uniformly over τ∈Υ, 
	
    √(n)(q̂(τ)-q(τ))ℬ_ipw(τ), as n→∞,

	where ℬ_ipw(·) is a scalar Gaussian process with covariance kernel Σ_ipw(·,·). The expression for Σ_ipw(·,·) can be found in the Appendix. 	

Two remarks are in order. First, the asymptotic variance for q̂(τ) is 

    ζ_Y^2(π,τ) + ζ_S^2(τ).

When strong balance is not achieved and the stratification is relevant, we have ζ_A^2(π,τ)>0. Thus, q̂(τ) is more efficient than β̂_1(τ) in the sense that

    Σ_ipw(τ,τ) < Σ_sqr(τ,τ).

When strong balance is achieved (γ(s) = 0), we have ζ_A^2(π,τ)=0. Thus, the two estimators are asymptotically first-order equivalent. Based on the same argument, one can potentially prove that, when strong balance is not achieved and the stratification is relevant, the IPW estimator for ATE has strictly smaller asymptotic variance than the simple two-sample difference and strata fixed effects estimators studied by <cit.>, and is asymptotically equivalent to the fully saturated linear regression estimator proposed by <cit.>. Second, since the amount of “balance" of the treatment assignment rule does not play a role in the limiting distribution of the IPW estimator, Assumption <ref>.3 is replaced by Assumption <ref>.4.  




§ WEIGHTED BOOTSTRAP 


In this section, we approximate the  asymptotic distributions of the SQR and IPW estimators via the weighted bootstrap. Let {ξ_i}_i=1^n be a sequence of bootstrap weights which will be specified later. Further denote 
n_1^w(s) = ∑_i =1^nξ_iA_i1{S_i=s}, n^w(s) = ∑_i =1^n ξ_i 1{S_i=s}, and π̂^w(s) = n_1^w(s)/n^w(s). The weighted bootstrap counterparts for the two estimators we study in this paper can then be written respectively as 

    β̂^w(τ) = _b ∑_i=1^n ξ_i ρ_τ(Y_i - Ȧ_i'b)

and 

    q̂^w(τ) = q̂_1^w(τ) - q̂_0^w(τ),

where 

    q̂_1^w(τ) = _q ∑_i =1^n ξ_iA_i/π̂^w(S_i)ρ_τ(Y_i - q)   and  q̂_0^w(τ) = _q ∑_i =1^n ξ_i(1-A_i)/1-π̂^w(S_i)ρ_τ(Y_i - q).

The second element β̂_1^w(τ) of β̂^w(τ) and q̂^w(τ) are the SQR and IPW bootstrap estimators for the τ-th QTE, respectively. Next, we specify the bootstrap weights. 

	
	Suppose {ξ_i}_i=1^n is a sequence of nonnegative i.i.d. random variables with unit expectation and variance and a sub-exponential upper tail. 	

The nonnegativity is required to maintain the convexity of the quantile regression objective function. The other conditions in Assumption <ref> are common for the weighted bootstrap approximation. In practice, we generate {ξ_i}_i=1^n by the standard exponential distribution. The corresponding weighted bootstrap is also known as the Bayesian bootstrap. 


	
	If Assumptions <ref>.1–<ref>.3, <ref>, and <ref> hold, then uniformly over τ∈Υ and conditionally on data,
	
    √(n)(β̂_1^w(τ) - β̂_1(τ)) ℬ̃_sqr(τ), as n→∞,

	where ℬ̃_sqr(τ) is a Gaussian process. In addition,  ℬ̃_sqr(τ) shares the same covariance kernel with ℬ_sqr(τ) defined in Theorems <ref> with γ(s) there replaced by π(1-π).
	
	If Assumptions <ref>.1, <ref>.2, <ref>.4, <ref>, <ref> hold,  then uniformly over τ∈Υ and conditionally on data, 	
	
    √(n)(q̂^w(τ) - q̂(τ)) ℬ_ipw(τ), as n→∞,

	where ℬ_ipw(τ) is the same Gaussian process defined in Theorem <ref>. 


Four remarks are in order. First, the weighted bootstrap sample does not preserve the negative cross-sectional dependence in the original sample. Asymptotic variances of the weighted bootstrap estimators equal those of their original sample counterparts as if SRS is applied. In fact, the asymptotic variance for β̂^w_1(τ) is

    ζ_Y^2(π,τ) +  ζ̃_A^2(π,τ) + ζ_S^2(τ),

where

    ζ̃_A^2(π,τ) = 𝔼π(1-π) (m_1(S,τ)/π f_1(q_1(τ)) + m_0(S,τ)/(1-π) f_0(q_0(τ)))^2.

This asymptotic variance is intuitive as the weight ξ_i is independent with each other, which implies that, conditionally on data, the bootstrap sample observations are independent. As γ(s) ≤π (1-π), we have 

    ζ_A^2(π,τ) ≤ζ̃_A^2(π,τ).

If the inequality is strict, then the weighted bootstrap overestimates the asymptotic variance of the SQR estimator, and thus, the Wald test constructed using the SQR estimator and its weighted bootstrap standard error is conservative.   

Second, the asymptotic distribution of the weighted bootstrap IPW estimator coincides with that of the original estimator. The asymptotic size of the Wald test constructed using the IPW estimator and its weighted bootstrap standard error then equals the nominal level. Theorem <ref> shows that the asymptotic variance for q̂(τ) is invariant in the treatment assignment rule applied. Thus, even though the weighted bootstrap sample ignores the cross-sectional dependence and behaves as if the treatment status is generated randomly, the asymptotic variance for q̂^w(τ) is still 

    ζ_Y^2(π,τ) + ζ_S^2(τ).
 

Third, the validity of weighted bootstrap for the IPW estimator only requires Assumption <ref>.4 instead of <ref>.3, for the same reason mentioned after Theorem <ref>. 


Fourth, it is possible to consider the conventional nonparametric bootstrap which generates the bootstrap sample from the empirical distribution of the data. If the observations are i.i.d., <cit.> show that the conventional bootstrap is first-order equivalent to a weighted bootstrap with Poisson(1) weights. However, in the current setting, {A_i}_i ≥ 1 is dependent. It is technically challenging to rigorously show that the above equivalence still holds. We leave it as an interesting topic for future research.  



§ COVARIATE-ADAPTIVE BOOTSTRAP


In this section, we consider the covariate-adaptive bootstrap procedure as follows:

	
  * Draw {S_i^*}_i=1^n from the empirical distribution of {S_i}_i=1^n with replacement. 
	
  * Generate {A_i^*}_i=1^n based on {S_i^*}_i=1^n and the treatment assignment rule. 
	
  * For A_i^* = a and S_i^* = s, draw Y_i^* from the empirical distribution of Y_i given A_i=a and S_i = s with replacement. 

First, Step 1 is the conventional nonparametric bootstrap. The bootstrap sample {S_i^*}_i=1^n is obtained by drawing from the empirical distribution of {S_i}_i=1^n with replacement n times. Second, Step 2 follows the treatment assignment rule, and thus preserves the cross-sectional dependence structure in the bootstrap sample, even after conditioning on data. The weighted bootstrap sample, by contrast, is cross-sectionally independent given data. Third, Step 3 applies the conventional bootstrap procedure to the outcome Y_i in the cell (S_i,A_i) = (s,a) ∈𝒮×{0,1}. Given that the original data contain n_a(s) observations in this cell, in this step, the bootstrap sample {Y_i^*}_i: A_i^* =a, S_i^*=s is obtained by drawing from the empirical distribution of these n_a(s) outcomes with replacement n^*_a(s) times, where n^*_a(s) = ∑_i=1^n 1{A_i^*=a, S_i^*=s}. Unlike the conventional bootstrap, here both n_a(s) and n_a^*(s) are random and are not necessarily the same. Last, to implement the covariate-adaptive bootstrap, researchers need to know the treatment assignment rule for the original sample. Unlike in observational studies, such information is usually available for RCTs. If one only knows that the treatment assignment rule achieves strong balance, then Theorem <ref> below still holds, provided that the bootstrap sample is generated from any treatment assignment rule that achieves strong balance. Even worse, if no information on the treatment assignment rule is available, then one cannot implement the covariate-adaptive bootstrap inference. In this case, the weighted bootstrap for the IPW estimator can still provide a non-conservative Wald test, as shown in Theorem <ref>.  



Using the bootstrap sample {Y_i^*,A_i^*,S_i^*}_i=1^n, we can estimate QTE by the two methods considered in the paper. Let 
n_1^*(s) = ∑_i =1^nA_i^*1{S_i^*=s}, n^*(s) = ∑_i =1^n 1{S_i^*=s}, π̂^*(s) = n_1^*(s)/n^*(s), and Ȧ_i^* = (1,A_i^*)'. Then, the two bootstrap estimators can be written respectively as 

    β̂^*(τ) = _b ∑_i=1^nρ_τ(Y_i^* - Ȧ_i^*b)

and 

    q̂^*(τ) = q̂_1^*(τ) - q̂_0^*(τ),

where 

    q̂_1^* = _q ∑_i =1^n A_i^*/π̂^*(S_i^*)ρ_τ(Y_i^* - q)   and  q̂_0^* = _q ∑_i =1^n 1-A_i^*/1-π̂^*(S_i^*)ρ_τ(Y_i^* - q).

The second element β̂_1^*(τ) of β̂^*(τ) and q̂^*(τ) are the SQR and IPW bootstrap estimators for the τ-th QTE, respectively. Parallel to Assumption <ref>, we make the following assumption for the bootstrap sample. 

	
	Let D_n^*(s) = ∑_i=1^n (A_i^* - π)1{S_i^* = s}. 
	
		
  * {{D^*_n(s)/√(n)}_s ∈𝒮|{S_i^*}_i=1^n  } N(0,Σ_D) a.s., where Σ_D = {p(s)γ(s):s ∈𝒮}.
		
  * sup_s ∈𝒮|D^*_n(s)|/√(n^*(s)) = O_p(1), sup_s ∈𝒮|D_n(s)|/√(n(s)) = O_p(1).
	

Assumption <ref>.1 is a high-level assumption. Obviously, it holds for SRS. For WEI, this condition holds by the same argument in <cit.> with the fact that n^*(s)/n(s) 1. For BCD, as shown in <cit.>, 

    D_n^*(s) |{S_i^*}_i=1^n = O_p(1).

Therefore, D_n^*(s)/√(n^*(s)) 0 and Assumption <ref>.1 holds with γ(s) = 0. For SBR, it is clear that |D_n^*(s)| ≤ 1. Thus, Assumption <ref>.1 holds with γ(s) = 0 as well. In addition, as p(s)>0, based on the standard bootstrap results, we have n^*(s)/n  p(s) and n(s)/n  p(s). Therefore, Assumption <ref>.1 is sufficient for Assumption <ref>.2. Last, note that Assumption <ref>.2 implies Assumption <ref>.4. 

	
	Suppose Assumptions <ref>.1, <ref>.2, <ref>, and <ref>.2 hold. Then, uniformly over τ∈Υ and conditionally on data,  
	
    √(n)(q̂^*(τ) - q̂(τ)) ℬ_ipw(τ),  as n→∞.

	If, in addition, Assumptions <ref>.3 and <ref>.1 hold, then 
	
    √(n)(β̂_1^*(τ) - q̂(τ)) ℬ_sqr(τ), as n→∞.

	Here, ℬ_sqr(τ) and ℬ_ipw(τ) are two Gaussian processes defined in Theorem <ref> and <ref>, respectively. 





Several remarks are in order. First, unlike the usual bootstrap estimator, the covariate-adaptive bootstrap SQR estimator is not centered around its corresponding counterpart from the original sample, but rather q̂(τ). The reason is that the treatment status A_i^* is not generated by bootstrap. In the linear expansion for the bootstrap estimator β̂_1^*(τ), the part of the influence function that accounts for the variation generated by A_i^* need not be centered. We also know from the proof of Theorem <ref> that q̂(τ) do not have an influence function that represents the variation generated by A_i. Thus, q̂(τ) can be used to center β̂_1^*(τ). 

Second, the choice of q̂(τ) as the center is somehow ad-hoc. In fact, any estimator q̃(τ) that is first-order equivalent to q̂(τ) in the sense that  

    sup_τ∈Υ|q̃(τ) - q̂(τ)| = o_p(1/√(n))
 
can serve as the center for the bootstrap estimators q̂^*(τ) and β̂_1^*(τ). 


Third, when the treatment assignment rule achieves strong balance, β̂_1(τ) and q̂(τ) are first-order equivalent. In this case, β̂_1(τ) can serve as the center for β̂_1^*(τ) and various bootstrap inference methods are valid. On the other hand, when the treatment assignment rule does not achieve strong balance, β̂_1(τ) and q̂(τ) are not first-order equivalent. In this case, the asymptotic size of the percentile bootstrap for the SQR estimator using the quantiles of β̂_1^*(τ) does not equal the nominal level. In the next section, we propose a way to compute the bootstrap standard error which does not depend on the choice of the center. Based on the bootstrap standard error, researchers can construct t-statistics and use standard normal critical values for inference.



Fourth, for ATE, we can use the same bootstrap sample to compute the standard errors for the simple and strata fixed effects estimators proposed in <cit.> as well as the IPW estimator. We expect that all the results in this paper hold for the ATE as well. 




§ SIMULATION


We can summarize four bootstrap scenarios from the analysis in Sections <ref> and <ref>: (1) the SQR estimator with the weighted bootstrap, (2) the IPW estimator with either the weighted or covariate-adaptive bootstrap, (3) the SQR estimator with the covariate-adaptive bootstrap when the assignment rule achieves strong balance, and (4) the SQR estimator with the covariate-adaptive bootstrap when the assignment rule does not achieve strong balance. The results of Sections <ref> and <ref> imply that the bootstrap in scenario (1) produces conservative Wald-tests when the treatment assignment rule is not SRS. For scenarios (2) and (3), various bootstrap based inference methods are valid. However,  for scenario (4), researchers should be careful about the centering issue. In particular, the percentile bootstrap inference using the quantiles of β̂_1^* is invalid.  In the following, we propose one single bootstrap inference method that works for scenarios (2)–(4). In addition, the proposed method does not require the knowledge of the centering. 

We take the IPW estimator as an example. We can repeat the bootstrap estimation[For the IPW estimator, we can use either the weighted or covariate-adaptive bootstrap. For the SQR estimator, we can only use the covariate-adaptive bootstrap.] B times and obtain B bootstrap IPW estimates, denoted as {q̂_b^*(τ)}_b=1^B. Further denote Q̂(α) as the α-th empirical quantile of {q̂_b^*(τ)}_b=1^B. We can test the null hypothesis that q(τ) = q^0(τ) via 1{|q̂(τ) - q^0(τ)/σ̂_n^*| > z_1-α/2}, where q̂(τ), z_1-α/2, and σ̂_n^* are the IPW estimator, the (1-α/2)-th quantile of the standard normal distribution, and 

    σ̂_n^* = Q̂(0.975) - Q̂(0.025)/z_0.975 - z_0.025,
  
respectively. In scenarios (2)–(4), the asymptotic size of such test equals the nominal level α. In scenarios (2) and (3), we recommend the t-statistic and confidence interval using this particular bootstrap standard error (i.e., σ̂_n^*) over other bootstrap inference methods (e.g., bootstrap confidence interval, percentile bootstrap confidence interval, etc.) because based on unreported simulations, they have better finite sample performance. 




 §.§ Data Generating Processes

We consider two DGPs with parameters γ =4, σ =2, and μ which will be specified later. 

	
  * Let Z be standardized Beta(2,2) distributed, S_i = ∑_j=1^41{Z_i ≤ g_j}, and (g_1,⋯,g_4) = (-0.25√(20),0,0.25√(20),0.5√(20)). The outcome equation is 
	
    Y_i = A_i μ + γ Z_i + η_i,
 
	where η_i = σ A_i _i,1 + (1-A_i)_i,2 and (_i,1,_i,2) are jointly standard normal. 
	
	
	
	
	
	
  * Let Z be uniformly distributed on [-2,2], S_i = ∑_j=1^41{Z_i ≤ g_j}, and (g_1,⋯,g_4) = (-1,0,1,2). The outcome equation is 
	
    Y_i = A_i μ+A_iν_i,1   + (1-A_i)ν_i,0 + η_i,

	where ν_i,0 = γ Z_i^2 1{|Z_i|≥ 1} + γ/4(2 - Z_i^2)1{|Z_i|<1}, ν_i,1 = -ν_i,0, η_i = σ(1+Z_i^2)A_i_i,1 + (1+Z_i^2)(1-A_i)_i,2, and (_i,1,_i,2) are mutually independent T(3)/3 distributed. 


When π = 1/2, for each DGP, we consider four randomization schemes:

	
  * SRS: Treatment assignment is generated as in Example <ref>.
	
  * WEI: Treatment assignment is generated as in Example <ref> with ϕ(x) = (1-x)/2.
	
  * BCD: Treatment assignment is generated as in Example <ref> with λ = 0.75. 
	
  * SBR: Treatment assignment is generated as in Example <ref>. 

When π≠ 0.5, BCD is not defined while WEI is not defined in the original paper (<cit.>). Recently, <cit.> generalizes the adaptive biased-coin design (i.e., WEI) to multiple treatment values and unequal target treatment ratios. Here, for π≠ 0.5, we only consider SRS and SBR as in <cit.>. We conduct the simulations with sample sizes n=200 and 400. The numbers of simulation replications and bootstrap samples are 1000. Under the null, μ=0 and we compute the true parameters of interest using simulations with 10^6 sample size and 10^4 replications. Under the alternative, we perturb the true values by μ = 1 and μ = 0.75 for n = 200 and 400, respectively. We report the results for the median QTE. Section <ref> contains additional simulation results for ATE and QTEs with τ = 0.25 and 0.75. All the observations made in this section still apply. 



 §.§ QTE, π = 0.5

We consider the Wald test with six t-statistics and 95% nominal rate. We construct the t-statistics using one of our two point estimates and some estimate of the standard error. We will reject the null hypothesis when the absolute value of the t-statistic is greater than 1.96. The details about the point estimates and standard errors are as follows: 

	
  * “s/naive": the point estimator is computed by the SQR and its standard error σ̂_naive(τ) is computed as 
	
    σ̂^2_naive(τ) =    τ(1-τ) - 1/n∑_i =1^nm̂^2_1(S_i,τ) /πf̂_1^2(q̂_1(τ))+τ(1-τ) - 1/n∑_i =1^nm̂^2_0(S_i,τ) /(1-π)f̂_0^2(q̂_0(τ))
        + 1/n∑_i =1^n π(1-π)(m̂_1(S_i,τ)/πf̂_1(q̂_1(τ)) + m̂_0(S_i,τ)/(1-π) f̂_0(q̂_0(τ)))^2 
        + 1/n∑_i =1^n (m̂_1(S_i,τ)/f̂_1(q̂_1(τ)) - m̂_0(S_i,τ)/f̂_0(q̂_0(τ)))^2,

	where q̂_j(τ) is the τ-th empirical quantile of Y_i|A_i = j, 
    m̂_i,1(s,τ) = ∑_i =1^nA_i1{S_i = s}(τ - 1{Y_i ≤q̂_1(τ)})/n_1(s),
 
	
    m̂_i,0(s,τ) = ∑_i =1^n(1-A_i)1{S_i = s}(τ - 1{Y_i ≤q̂_0(τ)})/n(s)-n_1(s).
 
	For j=0,1, f̂_j(·) is computed by the kernel density estimation using the observations Y_i provided that A_i = j, bandwidth h_j = 1.06σ̂_jn_j^-1/5, Gaussian kernel function, standard deviation σ̂_j of the observations Y_i provided that A_i = j, and n_j = ∑_i =1^n 1{A_i = j}. 
	
  * “s/adj": exactly the same as the “s/naive" method with one difference: replacing π(1-π) in (<ref>) by γ(S_i).
	
  * “s/W": the point estimator is computed by the SQR and its standard error σ̂_W(τ) is computed by the weighted bootstrap procedure. The bootstrap weights {ξ_i}_i=1^n are generated from the standard exponential distribution. Denote {β̂^w_1,b}_b=1^B as the collection of B weighted bootstrap SQR estimates. Then, 
	
    σ̂_W(τ) = Q̂(0.975) - Q̂(0.025)/z_0.975 - z_0.025,

	where Q̂(α) is the α-th empirical quantile of {β̂^w_1,b(τ)}_b=1^B. 
	
  * “ipw/W": the same as above with one difference: the estimation method for both the original and bootstrap samples is the IPW QR.  
	
  * “s/CA": the point estimator is computed by the SQR and its standard error σ̂_CA(τ) is computed by the covariate-adaptive bootstrap procedure. Denote {β̂^*_1,b}_b=1^B as the collection of B estimates obtained by the SQR applied to the samples generated by the covariate-adaptive bootstrap procedure. Then, 
	
    σ̂_CA(τ) = Q̂(0.975) - Q̂(0.025)/z_0.975 - z_0.025,

	where Q̂(α) is the α-th empirical quantile of {β̂^*_1,b(τ)}_b=1^B. 
	
  * “ipw/CA": the same as above with one difference:  the estimation method for both the original and bootstrap samples is the IPW QR.  


Tables <ref> and <ref> present the rejection probabilities (multiplied by 100) for the six t-tests under both the null hypothesis and the alternative hypothesis, with sample sizes n=200 and 400, respectively. In these two tables, columns M and A represent DGPs and treatment assignment rules, respectively. From the rejection probabilities under the null, we can make five observations. First, the naive t-test (“s/naive") is conservative for WEI, BCD, and SBR, which is consistent with the findings for ATE estimators by <cit.> and <cit.>. Second, although the asymptotic size of the adjusted t-test (“s/adj") is expected to equal the nominal level, it does not perform well for DGP2. The main reason is that, in order to analytically compute the standard error, we must compute nuisance parameters such as the unconditional densities of Y(0) and Y(1), which requires tuning parameters. We further compute the standard errors following (<ref>) with π(1-π) and the tuning parameter h_j replaced by γ(S_i) and 1.06C_fσ̂_jn_j^-1/5, respectively, for some constant C_f ∈ [0.5,1.5]. Figure <ref> plots the rejection probabilities of the “s/adj" t-tests against C_f for the BCD assignment rule with n=200, τ = 0.5, and π = 0.5. We see that (1) the rejection probability is sensitive to the choice of bandwidth, (2) there is no universal optimal bandwidth across two DGPs, and (3) the covariate-adaptive bootstrap t-tests (“s/CA") represented by the dotted dash lines are quite stable across different DGPs and close to the nominal rate of rejection. Third, the weighted bootstrap t-test for the SQR estimator (“s/W") is conservative, especially for the BCD and SBR assignment rules which achieve strong balance. Fourth, the rejection probabilities of the weighted bootstrap t-test for the IPW estimator (“ipw/W") are close to the nominal rate even for sample size n=200, which is consistent with Theorem <ref>. Last, the rejection rates for the two covariate-adaptive bootstrap t-tests (“s/CA" and “ipw/CA") are close to the nominal rate, which is consistent with Theorem <ref>.    










Turning to the rejection rates under the alternative in Tables <ref> and <ref>, we can make two additional observations. First, for BCD and SBR, the rejection probabilities (power) for “ipw/W", “s/CA", and “ipw/CA" are close. This is because both BCD and SBR achieve strong balance. In this case, the two estimators we propose are asymptotically first-order equivalent. Second, for DGP1 with SRS and WEI assignment rules, “ipw/CA" is more powerful than “s/CA". This confirms our theoretical finding that the IPW estimator is strictly more efficient than the SQR estimator when the treatment assignment rule does not achieve strong balance. For DGP2 the three t-tests, i.e., “ipw/W", “s/CA", and “ipw/CA", have similar power. 








 §.§ QTE, π = 0.7

Tables <ref> and <ref> show the similar results with π = 0.7. The same comments for Tables <ref> and <ref> still apply.  







 §.§ Difference between Two QTEs

Last, we consider to infer q(0.25)-q(0.75) when π=0.5:

    H_0: q(0.25)-q(0.75) = the true value  v.s.   H_1: q(0.25)-q(0.75) = the true value+μ,

where μ = 1 and 0.75 for sample sizes 200 and 400, respectively. The two estimators for QTEs at τ=0.25 and 0.75 are correlated. We can compute the naive and adjusted standard errors for the SQR estimator by taking this covariance structure into account.[The formulas for the covariances can be found in the proofs of Theorems <ref> and <ref>.] On the other hand, in addition to avoiding the tuning parameters, another advantage of the bootstrap inference is it does not require the knowledge of this complicated covariance structure. Researchers may construct the t-statistic using the difference of two QTE estimators with the corresponding weighted and covariate-adaptive bootstrap standard errors, which are calculated using the exact same procedure as in Sections <ref> and <ref>. Taking the SQR estimator as an example, we estimate q(0.25)-q(0.75) via β̂_1(0.25) - β̂_1(0.75) and the corresponding covariate-adaptive bootstrap standard error is 

    σ̂_CA = Q̂(0.975) - Q̂(0.025)/z_0.975 - z_0.025,

where  Q̂(α) is the α-th empirical quantile of {β̂^*_1,b(0.25) - β̂^*_1,b(0.75)}_b=1^B. 




Based on the rejection rates reported in Tables <ref> and <ref>, the general observations for the previous simulation results still apply. Although under the null, the rejection rates for “ipw/W", “S/CA", “ipw/CA" in DGP2 are below the nominal 5%, they gradually increase as the sample size increases from 200 to 400.  







§ GUIDANCE FOR PRACTITIONERS


We recommend employing the t-statistic (or equivalently, the confidence interval) constructed using the IPW estimator and its weighted bootstrap standard error for inference in covariate-adaptive randomization, for the following four reasons. First, its asymptotic size equals the nominal level. Second, the IPW estimator has a smaller asymptotic variance than the SQR estimator when the treatment assignment rule does not achieve strong balance and the stratification is relevant.[In this case, for ATE, the IPW estimator also has a strictly smaller asymptotic variance than the strata fixed effects estimator studied in <cit.>.] Third, compared with the covariate-adaptive bootstrap, the validity of the weighted bootstrap requires a weaker condition that sup_s ∈𝒮|D_n(s)/n(s)| = o_p(1). Fourth, this method does not require the knowledge of the exact treatment assignment rule, thus is suitable in settings where such information is lacking, e.g., using someone else's RCT or subsample analysis. When the treatment assignment rule achieves strong balance, SQR estimator can also be used. But in this case, only the covariate-adaptive bootstrap standard error is valid. Last, the Wald test using SQR estimator and the weighted bootstrap standard error is not recommended, as it is conservative when the treatment assignment rule introduces negative dependence (i.e., γ(s) < π(1-π)) such as WEI, BCD, and SBR. 





§ EMPIRICAL APPLICATION



We illustrate our methods by estimating and inferring the average and quantile treatment effects of iron efficiency on educational attainment. The dataset we use is the same as the one analyzed by <cit.> and <cit.>. 



 §.§ Data Description


The dataset consists of 215 students from one Peruvian secondary school during the 2009 school year. About two thirds of students were assigned to the treatment group (A=1 or A=2). The other one third of students were assigned to the control group (A=0). One half of the students in the treatment group were shown a video in which a physician encouraged iron supplements (A=1) and the other half were shown the same encouragement from a popular soccer player (A=2). Those assignments were stratified by the number of years of secondary school completed (𝒮={1,⋯,5}). The field experiment used a stratified block randomization scheme with fractions (1/3, 1/3, 1/3) for each group, which achieves strong balance (γ(s)=0). 

In the following, we focus on the observations with A=0 and A=1, and estimate the treatment effect of the exposure to a video of encouraging iron supplements by a physician only. This practice was also implemented in <cit.>. In this case, the target proportions of treatment is π = 1/2. As in <cit.>, it is also possible to combine the two treatment groups, i.e., A=1 and A=2 and compute the treatment effects of exposure to a video of encouraging iron supplements by either a physician or a popular soccer player. Last, one can use the method developed in <cit.> to estimate the ATEs under multiple treatment status. However, in this setting, the estimation of QTE and the validity of bootstrap inference have not been investigated yet and are interesting topics for future research. 


For each observation, we have three outcome variables: number of pills taken, grade point average, and cognitive ability measured by the average score across different Nintendo Wii games. For more details about the outcome variables, we refer interested readers to <cit.>. In the following, we focus on the grade point average only as the other two outcomes are discrete. 




 §.§ Computation


We consider three pairs of point estimates and their corresponding non-conservative standard errors: (1) the SQR estimator with the covariate-adaptive bootstrap standard error, (2) the IPW estimator with the covariate-adaptive bootstrap standard error, and (3) the IPW estimator with the weighted bootstrap standard error. We denote them as “s/CA", “ipw/CA", and “ipw/W", respectively. For comparison, we also compute the SQR estimator with its  weighted bootstrap standard error, which is denoted as “s/W" . The SQR estimator for the τ-th QTE refers to β̂_1(τ) as the second element of β̂(τ) = (β̂_0(τ),β̂_1(τ)), where 

    β̂(τ) = _b = (b_0,b_1)^'∈^2∑_i=1^nρ_τ(Y_i - Ȧ_i'b),

Ȧ_i = (1,A_i)^', and ρ_τ(u) = u(τ - 1{u≤ 0}) is the standard check function. It is also just the difference between the τ-th empirical quantiles of treatment and control groups. The IPW estimator refers to q̂(τ) = q̂_1(τ) - q̂_0(τ), where

    q̂_1(τ) = _q 1/n∑_i=1^nA_i/π̂(S_i)ρ_τ(Y_i - q),   q̂_0(τ) = _q1/n∑_i=1^n 1-A_i/1-π̂(S_i)ρ_τ(Y_i - q),

π̂(·) denotes the propensity score estimator, π̂(s) = n_1(s)/n(s), n_1(s) = ∑_i=1^n A_i 1{S_i = s}, and n(s) = ∑_i=1^n 1{S_i = s}. The covariate-adaptive bootstrap standard error (“CA") refers to the standard error computed in Section <ref>. In particular, we can draw the covariate-adaptive bootstrap sample (Y_i^*,A_i^*,S_i^*)_i=1^n following the procedure in Section <ref>. We then recompute the SQR and IPW estimates using the bootstrap sample. We repeat the bootstrap estimation B times, and obtain 
{β̂_b,1^*(τ),q̂_b^*(τ)}_b=1^B. The standard errors for SQR and IPW estimates are computed as 

    σ̂_sqr(τ) = Q̂_sqr(0.975) - Q̂_sqr(0.025)/z_0.975 - z_0.025  and  σ̂_ipw(τ) = Q̂_ipw(0.975) - Q̂_ipw(0.025)/z_0.975 - z_0.025,

respectively, where Q̂_sqr(α) and Q̂_ipw(α) are the α-th empirical quantiles of {β̂_b,1^*(τ)}_b=1^B and {q̂_b^*(τ)}_b=1^B, respectively, and z_α is the α-th percentile of the standard normal distribution, i.e., z_0.975≈ 1.96 and z_0.025≈ -1.96. The weighted bootstrap standard error for the IPW estimate can be computed in the same manner with only one difference, the covariate-adaptive bootstrap estimator {q̂_b^*(τ)}_b=1^B is replaced by the weighted bootstrap estimator {q̂_b^w(τ)}_b=1^B, where for the b-th replication,  q̂_b^w(τ) = q̂_b,1^w(τ) - q̂_b,0^w(τ),

    q̂_b,1^w(τ) = _q 1/n∑_i=1^nξ_i^bA_i/π̂^w(S_i)ρ_τ(Y_i - q), q̂_b,1^w(τ) = _q1/n∑_i=1^n ξ_i^b(1-A_i)/1-π̂^w(S_i)ρ_τ(Y_i - q),

{ξ_i^b}_i=1^n is a sequence of i.i.d. standard exponentially distributed random variables, π̂^w(s) = n_1^w(s)/n^w(s), n_1^w(s) = ∑_i =1^nξ_iA_i1{S_i=s}, and n^w(s) = ∑_i =1^n ξ_i 1{S_i=s}. Similarly, we compute the weighted bootstrap SQR estimates {β_b,1^w(τ)}_b=1^B as the second element of {β_b^w(τ)}_b=1^B, where 

    β_b^w(τ) = _b = (b_0,b_1)^'∈^21/n∑_i=1^n ξ_i^bρ_τ(Y_i - Ȧ_i'b).





For the ATEs, we also compute the SQR estimator with the adjusted standard error based on the analytical formula derived by <cit.>, i.e., “s/adj". For QTE estimates, we consider quantile indexes {0.1,0.15,⋯,0.90}. The number of replications for the two bootstrap methods is B=1000.   







 §.§ Main Results

Table <ref> shows the estimates with the corresponding standard errors in parentheses. From the table, we can make several remarks. First, for both ATE and QTE, the SQR and IPW estimates are very close to each other and so do their standard errors computed via the analytical formula, weighted bootstrap, and covariate-adaptive bootstrap. This is consistent with our theory that, under strong balance, the two estimators are first-order equivalent. Second, although in theory, the weighted bootstrap standard errors for the SQR estimators should be larger than those computed via the covariate-adaptive bootstrap, in this application, they are very close. This is consistent with the finding in <cit.> that their adjusted p-value for the ATE estimate is close to the naive one. It implies the stratification may be irrelevant for the full-sample analysis. Third, we do not compute the adjusted standard error for the QTEs as it requires tuning parameters. Fourth, the QTEs provide us a new insight that the impact of supplementation on grade promotion is only significantly positive at 25% among the three quantiles. This may imply that the policy of reducing iron deficits is more effective for lower-ranked students.


In order to provide more details on the QTE estimates, we plot the 95% point-wise confidence band in Figure <ref> with quantile index ranging from 0.1 to 0.9. The solid line and the shadow area represent the point estimate and its 95% point-wise confidence interval, respectively. The confidence interval is constructed by 
    [β̂-1.96σ̂(β̂),β̂+1.96σ̂(β̂)],

where β̂ and σ̂(β̂) are the point estimates and the corresponding standard errors described above. As we expected, all the four findings look the same and the estimates are only significantly positive at low quantiles (15%–30%). 





 §.§ Subsample Results


Following <cit.>, we further split the sample into two based on whether the student is anemic, i.e., Anem_i = 0 or 1. We anticipate that there is no treatment effect for the nonanemic students and positive effects for anemic ones. In this subsample analysis, the covariate-adaptive bootstrap is infeasible, as in each sub-group, the strong-balance condition may be lost and the treatment assignment rule is not necessarily SBR and is generally unknown.[As the anonymous referee pointed out, it is possible to implement the covariate-adaptive bootstrap on the full sample and pick out the observations in the subsample to construct a bootstrap subsample. The analysis can then be repeated on this covariate-adaptive bootstrap subsample. Establishing the validity of this procedure is left as a topic for future research. ] However, the weighted bootstrap is still feasible as it does not require the knowledge of the treatment assignment rule. According to Theorem <ref>, the IPW estimator paired with the weighted bootstrap standard error is valid if

    sup_s∈𝒮|D_n^(1)(s)/n^(1)(s)|≡   sup_s∈𝒮|∑_i =1^n(A_i-π)1{S_i=s}1{Anem_i=1}/∑_i =1^n1{S_i=s}1{Anem_i=1}|=o_p(1)

and 

    sup_s∈𝒮|D_n^(0)(s)/n^(0)(s)|≡   sup_s∈𝒮|∑_i =1^n(A_i-π)1{S_i=s}1{Anem_i=0}/∑_i =1^n1{S_i=s}1{Anem_i=0}|=o_p(1).

We maintain this mild condition in this section. In our sample, 

    sup_s∈𝒮|D_n^(1)(s)/n^(1)(s)| = 0   and  sup_s∈𝒮|D_n^(0)(s)/n^(0)(s)| = 0.071,
  
which indicate that (<ref>) and (<ref>) are plausible. 



From Table <ref> and Figure <ref>, we see that the QTE estimates are significantly positive for the anemic students when the quantile index is between around 20%–75%, but are insignificant for nonanemic students. The lack of significance at very low and high quantiles for the anemic subsample may be due to a poor asymptotic normal approximation at extreme quantiles. To extend the inference of extremal QTEs in <cit.> to the context of covariate-adaptive randomization is an interesting topic for future research.  We also note that for both subsamples, the weighted bootstrap standard errors for the SQR estimators are larger than those for the IPW estimators, which is consistent with Theorem <ref>. It implies, for both sub-groups, the stratification is relevant.  









§ CONCLUSION


This paper studies the estimation and bootstrap inference for QTEs under covariate-adaptive randomization. We show that the weighted bootstrap standard error is only valid for the IPW estimator while the covariate-adaptive bootstrap standard error is valid for both SQR and IPW estimators. In the empirical application, we find that the QTE of iron supplementation on grade promotion is trivial for nonanemic students, while the impact is significantly positive for middle-ranked anemic students.




	Supplement to “Quantile Treatment Effects and Bootstrap Inference under Covariate-Adaptive Randomization"



This paper  gathers the supplementary material to the original paper. Sections <ref>, <ref>, <ref>, and <ref> contain the proofs for Theorems <ref>, <ref>, <ref>, and <ref>, respectively. Section <ref> contains the proofs for the technical lemmas. A separate supplement contains the analysis of strata fixed effects quantile regression estimator as well as additional simulation results.

  

 





§ PROOF OF THEOREM <REF>


Let u=(u_0, u_1)' ∈^2 and 

    L_n(u,τ)  = ∑_i=1^n[ρ_τ(Y_i - Ȧ_i'β(τ) - Ȧ_i'u/√(n)) - ρ_τ(Y_i - Ȧ_i'β(τ))].
  
Then, by the change of variable, we have that 

    √(n)(β̂(τ) - β(τ)) = _u L_n(u,τ).

Notice that L_n(u,τ) is convex in u for each τ and bounded in τ for each u. In the following, we aim to show that there exists 

    g_n(u,τ) = - u'W_n(τ) + 1/2u'Q(τ)u
 
such that (1) for each u, 

    sup_τ∈Υ|L_n(u,τ) - g_n(u,τ)|  0;

(2) the maximum eigenvalue of Q(τ) is bounded from above and the minimum eigenvalue of Q(τ) is bounded away from 0, uniformly over τ∈Υ; (3) W_n(τ) ℬ̃(τ) uniformly over τ∈Υ, in which ℬ̃(·) is some Gaussian process. Then by <cit.>, we have 

    √(n)(β̂(τ) - β(τ)) = [Q(τ)]^-1W_n(τ) + r_n(τ),

where sup_τ∈Υ||r_n(τ)|| = o_p(1). In addition, by (3), we have, uniformly over τ∈Υ, 

    √(n)(β̂(τ) - β(τ))  [Q(τ)]^-1ℬ̃(τ) ≡ℬ(τ).

The second element of ℬ(τ) is ℬ_sqr(τ) stated in Theorem <ref>. In the following, we prove requirements (1)–(3) in three steps. 

Step 1. By Knight's identity (<cit.>), we have 

    L_n(u,τ) 
    
    =     -u'∑_i=1^n1/√(n)Ȧ_i(τ- 1{Y_i≤Ȧ_i^'β(τ)}) + ∑_i=1^n∫_0^Ȧ_iu/√(n)(1{Y_i -  Ȧ_i^'β(τ)≤ v} - 1{Y_i -  Ȧ_i^'β(τ)≤ 0})dv 
    ≡    -u'W_n(τ) +Q_n(u,τ),

where 

    W_n(τ) = ∑_i=1^n1/√(n)Ȧ_i(τ- 1{Y_i≤Ȧ_i^'β(τ)})

and 

    Q_n(u,τ) =    ∑_i=1^n∫_0^Ȧ_i^' u/√(n)(1{Y_i -  Ȧ_i^'β(τ)≤ v} - 1{Y_i -  Ȧ_i^'β(τ)≤ 0})dv 
    
    =    ∑_i=1^n A_i∫_0^u_0+u_1/√(n)(1{Y_i(1) - q_1(τ)≤ v} - 1{Y_i(1) - q_1(τ)≤ 0})dv 
        + ∑_i=1^n (1-A_i)∫_0^u_0/√(n)(1{Y_i(0) - q_0(τ)≤ v} - 1{Y_i(0) - q_0(τ)≤ 0})dv 
    ≡    Q_n,1(u,τ) + Q_n,0(u,τ).

We first consider Q_n,1(u,τ). Following <cit.>, we define {(Y_i^s(1),Y_i^s(0)): 1≤ i ≤ n} as a sequence of i.i.d. random variables with marginal distributions equal to the distribution of (Y_i(1),Y_i(0))|S_i = s. The distribution of Q_n,1(u,τ) is the same as the counterpart with units ordered by strata and then ordered by A_i = 1 first and A_i = 0 second within each stratum, i.e., 

    Q_n,1(u,τ) d=   ∑_s ∈𝒮∑_i=N(s)+1^N(s)+n_1(s)∫_0^u_0+u_1/√(n)(1{Y_i^s(1) - q_1(τ)≤ v} - 1{Y_i^s(1) - q_1(τ)≤ 0})dv 
    
    =    ∑_s ∈𝒮[Γ^s_n(N(s) + n_1(s),τ) - Γ^s_n(N(s),τ) ],
 
where N(s) = ∑_i=1^n 1{S_i < s}, n_1(s) = ∑_i=1^n 1{S_i = s}A_i, and

    Γ^s_n(k,τ) = ∑_i=1^k∫_0^u_0+u_1/√(n)(1{Y_i^s(1) - q_1(τ)≤ v} - 1{Y_i^s(1) - q_1(τ)≤ 0})dv.


In addition, note that 

    ℙ(sup_t ∈ (0,1),τ∈Υ|Γ^s_n(⌊ nt ⌋,τ) - 𝔼Γ^s_n(⌊ nt ⌋,τ)|> ) 
    
    =    ℙ(max_1 ≤ k ≤ nsup_τ∈Υ|Γ^s_n(k,τ) - 𝔼Γ^s_n(k,τ)|> )  
    ≤    3 max_1 ≤ k ≤ nℙ(sup_τ∈Υ|Γ^s_n(k,τ) - 𝔼Γ^s_n(k,τ)|> /3) 
    ≤    9 ℙ(sup_τ∈Υ|Γ^s_n(n,τ) - 𝔼Γ^s_n(n,τ)|> /30) 
    ≤   270 𝔼sup_τ∈Υ|Γ^s_n(n,τ) - 𝔼Γ^s_n(n,τ)|/ = o(1).

The first inequality holds due to Lemma <ref> with S_k = Γ^s_n(k,τ) - 𝔼Γ^s_n(k,τ) and ||S_k|| = sup_τ∈Υ |Γ^s_n(k,τ) - 𝔼Γ^s_n(k,τ)|. The second inequality holds due to <cit.>. To derive the last equality of (<ref>), we consider the class of functions

    ℱ = {∫_0^u_0+u_1/√(n)(1{Y_i^s(1) - q_1(τ)≤ v} - 1{Y_i^s(1) - q_1(τ)≤ 0})dv: τ∈Υ}

with envelope |u_0+u_1|/√(n) and 

    sup_f ∈ℱ𝔼f^2 ≤sup_τ∈Υ𝔼[u_0+u_1/√(n)1{|Y_i^s(1) - q_1(τ)|≤u_0+u_1/√(n)}]^2 ≲ n^-3/2.

Note that ℱ is a VC-class with a fixed VC index. Therefore, by <cit.>, 

    𝔼sup_τ∈Υ|Γ^s_n(n,τ) - 𝔼Γ^s_n(n,τ)| = n||ℙ_n  - ℙ||_ℱ≲ n[√(log(n)/n^5/2) + log(n)/n^3/2] = o(1).
 
Then, (<ref>) implies that 

    sup_τ∈Υ|Q_n,1(u,τ) - ∑_s∈𝒮𝔼[Γ^s_n(⌊ n(N(s)/n + n_1(s)/n)⌋,τ) - Γ^s_n(⌊ n(N(s)/n)⌋,τ) ]| = o_p(1),

where following the convention in the empirical process literature,

    𝔼[Γ^s_n(⌊ n(N(s)/n + n_1(s)/n)⌋,τ) - Γ^s_n(⌊ n(N(s)/n)⌋,τ) ]

is interpreted as 

    𝔼[Γ^s_n(⌊ nt_2⌋,τ) - Γ^s_n(⌊ nt_1⌋,τ) ]_t_2 = N(s)/n,t_2 = N(s) + n_1(s)/n.
 
In addition,  N(s)/n  F(s) = F(S_i < s) and n_1(s)/n π p(s). Thus, uniformly over τ∈Υ, 

    𝔼[Γ^s_n(⌊ n(N(s)/n + n_1(s)/n)⌋,τ) - Γ^s_n(⌊ n(N(s)/n)⌋,τ) ] 
    
    =     n_1(s) ∫_0^u_0 + u_1/√(n)(F_1(q_1(τ)+v|s) - F_1(q_1(τ)|s))dv 
       π p(s) f_1(q_1(τ)|s)(u_0+u_1)^2/2,

where F_1(·|s) and f_1(·|s) are the conditional CDF and PDF of Y_1 given S=s, respectively. Then, uniformly over τ∈Υ,

    Q_n,1(u,τ) ∑_s ∈𝒮π p(s) f_1(q_1(τ)|s)(u_0+u_1)^2/2 = π f_1(q_1(τ))(u_0+u_1)^2/2.

Similarly, we can show that, uniformly over τ∈Υ, 

    Q_n,0(u,τ) (1-π) f_0(q_0(τ))u_0^2/2,

and thus

    Q_n(u,τ) 1/2u'Q(τ)u,

where 

    Q(τ) = [ π f_1(q_1(τ)) + (1-π)f_0(q_0(τ))                    π f_1(q_1(τ));                    π f_1(q_1(τ))                    π f_1(q_1(τ)) ].

Then, 

    sup_τ∈Υ|L_n(u,τ) - g_n(u,τ)| = sup_τ∈Υ|Q_n(u,τ) - 1/2u'Q(τ)u| = o_p(1).

This concludes the first step. 

Step 2. Note that det(Q(τ)) = π(1-π)f_1(q_1(τ))f_0(q_0(τ)), which is bounded and bounded away from zero. In addition, it can be shown that the two eigenvalues of Q are nonnegative. This leads to the desired result.

Step 3. Let e_1 = (1,1)^' and e_0 = (1,0)^'. Then, we have 

    W_n(τ) =    e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)A_i 1{S_i = s}(τ - 1{Y_i(1) ≤ q_1(τ)})
        + e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (1-A_i) 1{S_i = s}(τ - 1{Y_i(0) ≤ q_0(τ)}).

Let m_j(s,τ) = 𝔼(τ - 1{Y_i(j)≤ q_j(τ)}|S_i = s) and η_i,j(s,τ) = (τ - 1{Y_i(j) ≤ q_j(τ)}) - m_j(s,τ), j = 0,1. Then, 

    W_n(τ) =    [e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)A_i 1{S_i = s}η_i,1(s,τ) + e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (1-A_i) 1{S_i = s}η_i,0(s,τ)] 
        + [e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)(A_i-π) 1{S_i = s}m_1(s,τ) - e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (A_i-π) 1{S_i = s}m_0(s,τ)] 
        + [e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)π 1{S_i = s}m_1(s,τ) + e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (1-π) 1{S_i = s}m_0(s,τ) ] 
    ≡     W_n,1(τ) + W_n,2(τ) + W_n,3(τ).

By Lemma <ref>, uniformly over τ∈Υ, 

    (W_n,1(τ), W_n,2(τ), W_n,3(τ))  (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)),

where (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)) are three independent two-dimensional Gaussian processes with covariance kernels Σ_1(τ_1,τ_2), Σ_2(τ_1,τ_2), and Σ_3(τ_1,τ_2), respectively. Therefore, uniformly over τ∈Υ,

    W_n(τ) ℬ̃(τ),

where ℬ̃(τ) is a two-dimensional Gaussian process with covariance kernel

    Σ̃(τ_1,τ_2) = ∑_j=1^3Σ_j(τ_1,τ_2).

Consequently, 

    √(n)(β̂(τ) - β(τ))  [Q(τ)]^-1ℬ̃(τ) ≡ℬ(τ),

where ℬ(τ) is a two-dimensional Gaussian process with covariance kernel

    Σ(τ_1,τ_2) =     [Q(τ_1)]^-1Σ̃(τ_1,τ_2)  [Q(τ_2)]^-1
    
    =    1/π f_1(q_1(τ_1))f_1(q_1(τ_2))[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)][ 0 0; 0 1 ]
        + 1/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)][  1 -1; -1  1 ]
        + ∑_s ∈𝒮p(s)γ(s)[m_1(s,τ_1)m_1(s,τ_2)/π^2 f_1(q_1(τ_1))f_1(q_1(τ_2))[ 0 0; 0 1 ] - m_1(s,τ_1)m_0(s,τ_2)/π(1-π) f_1(q_1(τ_1))f_0(q_0(τ_2))[  0  0;  1 -1 ]
        - m_0(s,τ_1)m_1(s,τ_2)/π(1-π) f_0(q_0(τ_1))f_1(q_1(τ_2))[  0  1;  0 -1 ] + m_0(s,τ_1)m_0(s,τ_2)/(1-π)^2 f_0(q_0(τ_1))f_0(q_0(τ_2))[  1 -1; -1  1 ]] 
        + 𝔼m_1(S,τ_1)m_1(S,τ_2)/f_1(q_1(τ_1))f_1(q_1(τ_2))[ 0 0; 0 1 ] + 𝔼m_1(S,τ_1)m_0(S,τ_2)/f_1(q_1(τ_1))f_0(q_0(τ_2))[  0  0;  1 -1 ]
        + 𝔼m_0(S,τ_1)m_1(S,τ_2)/f_0(q_0(τ_1))f_1(q_1(τ_2))[  0  1;  0 -1 ]+𝔼m_0(S,τ_1)m_0(S,τ_2)/f_0(q_0(τ_1))f_0(q_0(τ_2))[  1 -1; -1  1 ].


Focusing on the (2,2)-element of Σ(τ_1,τ_2), we can conclude that 

    √(n)(β̂_1(τ) - q(τ)) ℬ_sqr(τ),

where the Gaussian process ℬ_sqr(τ) has a covariance kernel 

    Σ_sqr(τ_1,τ_2) 
    
    =    min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)/π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))
        + 𝔼γ(S)[m_1(S,τ_1)m_1(S,τ_2)/π^2 f_1(q_1(τ_1))f_1(q_1(τ_2))+m_1(S,τ_1)m_0(S,τ_2)/π(1-π) f_1(q_1(τ_1))f_0(q_0(τ_2))
       + m_0(S,τ_1)m_1(S,τ_2)/π(1-π) f_0(q_0(τ_1))f_1(q_1(τ_2)) +  m_0(S,τ_1)m_0(S,τ_2)/(1-π)^2 f_0(q_0(τ_1))f_0(q_0(τ_2))] 
        +𝔼[m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1))] [m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))].









§ PROOF OF THEOREM <REF>


By Knight's identity, we have 

    √(n)(q̂_1(τ) - q_1(τ)) =    _u L_n(u,τ),

where 

    L_n(u,τ) ≡   ∑_i =1^n A_i/π̂(S_i)[ρ_τ(Y_i - q_1(τ) - u/√(n)) - ρ_τ(Y_i - q_1(τ))] 
    
    =     - L_1,n(τ)u + L_2,n(u,τ),


    L_1,n(τ) = 1/√(n)∑_i=1^n A_i/π̂(S_i)(τ - 1{Y_i ≤ q_1(τ)})

and 

    L_2,n(u,τ) = ∑_i=1^n A_i/π̂(S_i)∫_0^u/√(n)(1{Y_i ≤ q_1(τ)+v} - 1{Y_i ≤ q_1(τ)})dv.

We aim to show that there exists 

    g_ipw,n(u,τ) = - W_ipw,n(τ) u + 1/2Q_ipw(τ)u^2

such that (1) for each u, 

    sup_τ∈Υ|L_n(u,τ) - g_ipw,n(u,τ)| 0;

(2) Q_ipw(τ) is bounded and bounded away from zero uniformly over τ∈Υ. In addition, as a corollary of claim (3) below, sup_τ∈Υ|W_ipw,1,n(τ)| = O_p(1). Therefore, by <cit.>, we have 

    √(n)(q̂_1(τ) - q_1(τ)) = Q_ipw,1^-1(τ)W_ipw,1,n(τ) + R_ipw,1,n(τ),

where sup_τ∈Υ|R_ipw,1,n(τ)| = o_p(1). Similarly, we can show that 

    √(n)(q̂_0(τ) - q_0(τ)) = Q_ipw,0^-1(τ)W_ipw,0,n(τ) + R_ipw,0,n(τ),
 
where sup_τ∈Υ|R_ipw,0,n(τ)| = o_p(1). Then, 

    √(n)(q̂(τ) - q(τ)) =  Q_ipw,1^-1(τ)W_ipw,1,n(τ) - Q_ipw,0^-1(τ)W_ipw,0,n(τ) + R_ipw,1,n(τ) - R_ipw,0,n(τ).

Last, we aim to show that, (3) uniformly over τ∈Υ, 

    Q_ipw,1^-1(τ)W_ipw,1,n(τ) - Q_ipw,0^-1(τ)W_ipw,0,n(τ) ℬ_ipw(τ),

where ℬ_ipw(τ) is a scalar Gaussian process with covariance kernel Σ_ipw(τ_1,τ_2). We prove claims (1)–(3) in three steps. 

Step 1. For L_1,n(τ), we have 

    L_1,n(τ) =    1/√(n)∑_i=1^n ∑_s ∈𝒮A_i/π1{S_i=s}(τ - 1{Y_i(1) ≤ q_1(τ)}) 
        - ∑_i=1^n ∑_s ∈𝒮A_i1{S_i = s}(π̂(s) - π)/√(n)π̂(s)π(τ - 1{Y_i(1) ≤ q_1(τ)}) 
    
    =    1/√(n)∑_i=1^n ∑_s ∈𝒮A_i/π1{S_i=s}(τ - 1{Y_i(1) ≤ q_1(τ)}) 
        - ∑_i=1^n ∑_s ∈𝒮A_i1{S_i = s}D_n(s)/n(s)√(n)π̂(s)πη_i,1(s,τ) -  ∑_s ∈𝒮D_n(s)m_1(s,τ)/n(s)√(n)π̂(s)π D_n(s) - ∑_s ∈𝒮D_n(s)m_1(s,τ)/√(n)π̂(s)
     
    =    ∑_s ∈𝒮1/√(n)∑_i =1^n A_i 1{S_i = s}/πη_i,1(s,τ) + ∑_s ∈𝒮D_n(s)/√(n)πm_1(s,τ) + ∑_i=1^n m_1(S_i,τ)/√(n)
        - ∑_i=1^n ∑_s ∈𝒮A_i1{S_i = s}D_n(s)/n(s)√(n)π̂(s)πη_i,1(s,τ) -  ∑_s ∈𝒮D_n(s)m_1(s,τ)/n(s)√(n)π̂(s)π D_n(s) - ∑_s ∈𝒮D_n(s)m_1(s,τ)/√(n)π̂(s)
    
    =     W_ipw,1,n(τ)+ R_ipw(τ),

where 

    W_ipw,1,n(τ) =  ∑_s ∈𝒮1/√(n)∑_i =1^n A_i 1{S_i = s}/πη_i,1(s,τ)  + ∑_i=1^n m_1(S_i,τ)/√(n)

and 

    R_ipw(τ) 
    
    =     - ∑_i=1^n ∑_s ∈𝒮A_i1{S_i = s}D_n(s)/n(s)√(n)π̂(s)πη_i,1(s,τ) -  ∑_s ∈𝒮D_n(s)m_1(s,τ)/n(s)√(n)π̂(s)π D_n(s) + ∑_s ∈𝒮D_n(s)m_1(s,τ)/√(n)(1/π - 1/π̂(s)) 
    
    =     - ∑_i=1^n ∑_s ∈𝒮A_i1{S_i = s}D_n(s)/n(s)√(n)π̂(s)πη_i,1(s,τ),

where we use the fact that π̂(s) - π = D_n(s)/n(s). By the same argument in Claim (1) of the proof of Lemma <ref>, we have, for every s∈𝒮,

    sup_τ∈Υ|1/√(n)∑_i =1^n A_i 1{S_i=s}η_i,1(s,τ)| d=sup_τ∈Υ|1/√(n)∑_i =N(s)+1^N(s)+n(s)η̃_i,1(s,τ)| = O_p(1),

where η̃_i,j(s,τ) = τ - 1{Y_i^s(j) ≤ q_j(τ) } - m_j(s,τ), for j = 0,1, where {Y_i^s(0),Y_i^s(1)}_i ≥ 1 are the same as defined in Step 1 in the proof of Theorem <ref>.



Because of (<ref>) and the fact that D_n(s)/n(s) = o_p(1), we have 

    sup_τ∈Υ|R_ipw(τ)| = o_p(1).

For L_2,n(u,τ), we have 

    L_2,n(u,τ) =    ∑_s ∈𝒮1/π̂(s)∑_i=N(s)+1^N(s)+n_1(s)∫_0^u/√(n)(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)+v})dv 
    
    =    ∑_s ∈𝒮1/π̂(s)[Γ_n^s(N(s)+n_1(s),τ) - Γ_n^s(N(s),τ) ],

where 

    Γ_n^s(k,τ) = ∑_i=1^k ∫_0^u/√(n)(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)+v})dv.

By the same argument in (<ref>), we can show that 

    sup_t ∈ (0,1),τ∈Υ|Γ_n^s(⌊ nt ⌋,τ) - 𝔼Γ_n^s(⌊ nt ⌋,τ)| = o_p(1).

In addition,  

    𝔼Γ_n^s(N(s)+n_1(s),τ) - 𝔼Γ_n^s(N(s),τ) π p(s)f_1(q_1(τ)|s)u^2/2.

Therefore, 

    sup_τ∈Υ|L_2,n(u,τ) - f_1(q_1(τ))u^2/2| = o_p(1),

where we use the fact that π̂(s)-π = D_n(s)/n(s) = o_p(1) and 

    ∑_s ∈𝒮p(s)f_1(q_1(τ)|s) = f_1(q_1(τ)).

This establishes (<ref>) with Q_ipw,1(τ) = f_1(q_1(τ)) and W_ipw,n(τ) defined in (<ref>).

Step 2. Statement (2) holds by Assumption <ref>. 

Step 3. By a similar argument in Step 1, we have 

    W_ipw,0,n(τ) = ∑_s ∈𝒮1/√(n)∑_i =1^n (1-A_i) 1{S_i = s}/1-πη_i,0(s,τ)  + ∑_i=1^n m_0(S_i,τ)/√(n)

and Q_ipw,0(τ) = f_0(q_0(τ)). Therefore, 

    √(n)(q̂ - q) =    1/√(n)∑_s ∈𝒮∑_i =1^n [A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] 
        + [1/√(n)∑_i=1^n(m_1(S_i,τ)/f_1(q_1(τ)) -  m_0(S_i,τ)/f_0(q_0(τ)))] + R_ipw,n(τ) 
    
    =    𝒲_n,1(τ)+𝒲_n,2(τ)+ R_ipw,n(τ)

where sup_τ∈Υ|R_ipw,n(τ)| = o_p(1). Last, Lemma <ref> establishes that 

    (𝒲_n,1(τ),𝒲_n,2(τ))  (ℬ_ipw,1(τ),ℬ_ipw,2(τ)),

where (ℬ_ipw,1(τ),ℬ_ipw,2(τ)) are two mutually independent scalar Gaussian processes with covariance kernels 

    Σ_ipw,1(τ_1,τ_2) = min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)  /π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π)f_0(q_0(τ_1))f_0(q_0(τ_2))

and 

    Σ_ipw,2(τ_1,τ_2) = 𝔼(m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1)))(m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))),

respectively. In particular, the asymptotic variance for q̂ is 

    ζ_Y^2(π,τ) + ζ_S^2(τ),

where ζ_Y^2(π,τ) and ζ_S^2(τ) are the same as those in the proof of Theorem <ref>. 




§ PROOF OF THEOREM <REF>


First, we consider the weighted bootstrap for the SQR estimator. Note that 

    √(n)(β̂^w(τ) - β(τ)) = _uL_n^w(u,τ),

where 

    L_n^w(u,τ) = ∑_i=1^n ξ_i[ρ_τ(Y_i - Ȧ_i'β(τ) - Ȧ_i'u/√(n)) - ρ_τ(Y_i - Ȧ_i'β(τ))].

Similar to the proof of Theorem <ref>, we can show that 

    sup_τ∈Υ|L_n^w(u,τ) - g_n^w(u,τ)| → 0,

where 

    g_n^w(u,τ) = - u'W_n^w(τ) + 1/2u'Q(τ)u,


    W_n^w(τ) = ∑_i=1^n ξ_i/√(n)Ȧ_i(τ - 1{Y_i ≤Ȧ'β(τ)}),

and Q(τ) is defined in (<ref>). Therefore, by <cit.>, we have 

    √(n)(β̂^w(τ) - β(τ)) = [Q(τ)]^-1W_n^w(τ) + r_n^w(τ),

where sup_τ∈Υ||r_n^w(τ)|| = o_p(1). By Theorem <ref>, 

    √(n)(β̂^w(τ) - β̂(τ)) =  [Q(τ)]^-1∑_i=1^n ξ_i-1/√(n)Ȧ_i(τ - 1{Y_i ≤Ȧ'β(τ)}) + o_p(1),

where the o_p(1) term holds uniformly over τ∈Υ. In addition, Lemma <ref> shows that, conditionally on data, the second element of 
[Q(τ)]^-1∑_i=1^n ξ_i-1/√(n)Ȧ_i(τ - 1{Y_i ≤Ȧ'β(τ)}) converges to ℬ̃_sqr(τ) uniformly over τ∈Υ. This leads to the desired result for the weighted bootstrap simple quantile regression estimator. 


Next, we turn to the IPW estimator. Denote q̂_j^w(τ), j=0,1 the weighted bootstrap counterpart of q̂_j(τ). We have

    √(n)(q̂_1^w(τ) - q_1(τ)) = _u L_n^w(u,τ),

where 

    L_n^w(u,τ) =    ∑_i =1^n ξ_iA_i/π̂^w(S_i)[ρ_τ(Y_i - q_1(τ) - u/√(n)) - ρ_τ(Y_i -q_1(τ))] 
    ≡    - L_1,n^w(τ) u + L_2,n^w(u,τ),

where 

    L_1,n^w(τ) = 1/√(n)∑_i =1^n ξ_iA_i/π̂^w(S_i)(τ - 1{Y_i ≤ q_1(τ)})

and

    L_2,n^w(τ) = ∑_i=1^n ξ_iA_i/π̂^w(S_i)∫_0^u/√(n)(1{Y_i ≤ q_1(τ)+v} - 1{Y_i ≤ q_1(τ)})dv.

Recall 

    D_n^w(s) = ∑_i =1^n ξ_i(A_i - π)1{S_i = s},    n^w(s) = ∑_i =1^n ξ_i1{S_i = s},
 
and 

    π̂^w(s) = ∑_i=1^n ξ_iA_i1{S_i=s}/n^w(s) = π + D_n^w(s)/n^w(s).
 
Then, for L_1,n^w(τ), we have 

    L_1,n^w(τ) =    1/√(n)∑_i=1^n ∑_s ∈𝒮ξ_iA_i/π1{S_i=s}(τ - 1{Y_i(1) ≤ q_1(τ)}) 
        - ∑_i=1^n ∑_s ∈𝒮ξ_iA_i1{S_i = s}(π̂^w(s) - π)/√(n)π̂^w(s)π(τ - 1{Y_i(1) ≤ q_1(τ)}) 
    
    =    1/√(n)∑_i=1^n ∑_s ∈𝒮ξ_iA_i/π1{S_i=s}(τ - 1{Y_i(1) ≤ q_1(τ)}) 
        - ∑_i=1^n ∑_s ∈𝒮ξ_iA_i1{S_i = s}D^w_n(s)/n^w(s)√(n)π̂(s)πη_i,1(s,τ) -  ∑_s ∈𝒮D^w_n(s)m_1(s,τ)/n^w(s)√(n)π̂^w(s)π D^w_n(s) - ∑_s ∈𝒮D^w_n(s)m_1(s,τ)/√(n)π̂^w(s)
     
    =    ∑_s ∈𝒮1/√(n)∑_i =1^n ξ_iA_i 1{S_i = s}/πη_i,1(s,τ) + ∑_s ∈𝒮D^w_n(s)/√(n)πm_1(s,τ) + ∑_i=1^n ξ_im_1(S_i,τ)/√(n)
        - ∑_s ∈𝒮D^w_n(s)∑_i=1^n ξ_iA_i1{S_i = s}/n^w(s)√(n)π̂^w(s)πη_i,1(s,τ) -  ∑_s ∈𝒮D^w_n(s)m_1(s,τ)/n^w(s)√(n)π̂^w(s)π D^w_n(s) - ∑_s ∈𝒮D^w_n(s)m_1(s,τ)/√(n)π̂^w(s)
    
    =     W_ipw,1,n^w(τ)+ R_ipw^w(τ),

where 

    W_ipw,1,n^w(τ) =  ∑_s ∈𝒮1/√(n)∑_i =1^n ξ_iA_i 1{S_i = s}/πη_i,1(s,τ)  + ∑_i=1^n ξ_im_1(S_i,τ)/√(n)

and 

    R_ipw^w(τ) 
    
    =     - ∑_s ∈𝒮D^w_n(s)∑_i=1^n ξ_iA_i1{S_i = s}/n^w(s)√(n)π̂^w(s)πη_i,1(s,τ) -  ∑_s ∈𝒮D^w_n(s)m_1(s,τ)/n^w(s)√(n)π̂^w(s)π D^w_n(s) + ∑_s ∈𝒮D^w_n(s)m_1(s,τ)/√(n)(1/π - 1/π̂^w(s)) 
    
    =     - ∑_s ∈𝒮D^w_n(s)∑_i=1^n ξ_iA_i1{S_i = s}/n^w(s)√(n)π̂^w(s)πη_i,1(s,τ).

In the following, we aim to show D_n^w(s)/n^w(s) = o_p(1) and 

    sup_τ∈Υ, s ∈𝒮|∑_i=1^nξ_iA_i1{S_i =s}η_i,1(s,τ)| = O_p(√(n)).

For the first claim, we note that n^w(s)/n(s)  1 and D_n(s)/n(s)  0. Therefore, we only need to show 

    D_n^w(s) - D_n(s)/n(s) = ∑_i=1^n(ξ_i-1)(A_i-π)1{S_i=s}/n(s) 0.

As n(s) →∞ a.s., given data, 


    1/n(s)∑_i=1^n(A_i - π)^21{S_i = s} =    1/n∑_i=1^n(A_i - π - 2π(A_i - π) + π - π^2)1{S_i=s}
    
    =    D_n(s) - 2π D_n(s)/n(s) + π(1-π) π (1-π).

Then, by the Lindeberg CLT, conditionally on data, 

    1/√(n(s))∑_i =1^n (ξ_i - 1)(A_i - π)1{S_i = s} N(0,π(1-π)) = O_p(1),

and thus 

    D_n^w(s) - D_n(s)/n(s) = O_p(n^-1/2(s)) = o_p(1).

This leads to the first claim. For the second claim, we note that 

    ∑_i=1^nξ_iA_i1{S_i =s}η_i,1(s,τ) = ∑_i = N(s)+1^N(s)+n_1(s)ξ_i η̃_i,1(s,τ).

We can show the RHS of the above display is O_p(√(n)) for all s ∈𝒮 following the same argument used in Claim (1) of the proof of Lemma <ref>. Given these two claims and by noticing that 

    π̂^w(s) - π = D^w_n(s)/n^w(s) = o_p(1),

we have 

    sup_τ∈Υ|R_ipw^w(τ)| = o_p(1).


Similar to the argument used to derive the limit of L_2,n(τ) in the proof of Theorem <ref>, we can show that 

    sup_τ∈Υ|L_2,n^w(u,τ) - f_1(q_1(τ))u^2/2| = o_p(1).

Therefore, 

    √(n)(q̂_1^w(τ) - q_1(τ)) = W_ipw,1,n^w(τ)/f_1(q_1(τ)) + R_1^w(τ),

where sup_τ∈Υ|R_1^w(τ)| = o_p(1). Similarly, 

    √(n)(q̂_0^w(τ) - q_0(τ)) = W_ipw,0,n^w(τ)/f_0(q_0(τ)) + R_0^w(τ),

where 

    W_ipw,0,n^w(τ) = ∑_s ∈𝒮1/√(n)∑_i =1^n ξ_i(1-A_i)1{S_i = s}/1-πη_i,0(s,τ) + ∑_i =1^nξ_im_0(S_i,τ)/√(n)

and sup_τ∈Υ|R_0^w(τ)| = o_p(1). Therefore, 

    √(n)(q̂^w(τ) - q̂(τ)) 
    
    =    ∑_s ∈𝒮1/√(n)∑_i =1^n (ξ_i-1){A_i1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ))-(1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π)f_0(q_0(τ))
        + [m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ))]1{S_i=s}} + o_p(1),

where the o_p(1) term holds uniformly over τ∈Υ. In order to show the conditional weak convergence, we only need to show the conditionally stochastic equicontinuity and finite-dimensional convergence. The former can be shown in the same manner as Lemma <ref>. For the latter, we note that 

    1/n∑_s ∈𝒮∑_i =1^n {A_i1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ))-(1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π)f_0(q_0(τ)) + [m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ))]1{S_i=s}}^2 
    
    =    ∑_s ∈𝒮1/n∑_i =1^n {(1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π)f_0(q_0(τ))}^2 + ∑_s ∈𝒮1/n∑_i =1^n {A_i1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ))}^2 
        + ∑_s ∈𝒮1/n∑_i =1^n  {[m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ))]1{S_i=s}}^2 
        + ∑_s ∈𝒮2/n∑_i =1^n {A_i1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ))}[m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ))] 
        - ∑_s ∈𝒮2/n∑_i =1^n{(1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π)f_0(q_0(τ))}[m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ))] 
       ζ_Y^2(π,τ) + ζ_S^2(τ).

Note that the RHS of the above display is the same as the asymptotic variance of the original estimator q̂(τ). By the CLT conditional on data, we can establish the one-dimensional weak convergence. Then, by the Cramér-Wold Theorem, we can extend such result to any finite dimension. This concludes the proof. 



§ PROOF OF THEOREM <REF>


It suffices to prove the theorem with 

    q̃(τ)  =       q(τ)+[∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋η̃_i,1(s,τ)/nπ f_1(q_1(τ)) -  ∑_s ∈𝒮∑_i = ⌊ n (F(s)+π p(s)) ⌋+1^⌊ n (F(s)+ p(s)) ⌋η̃_i,0(s,τ)/n(1-π)f_0(q_0(τ))] 
        + [∑_i=1^n 1/n(m_1(S_i,τ) /f_1(q_1(τ)) - m_0(S_i,τ) /f_0(q_0(τ))) ],

as we have shown in Theorem <ref> that 

    sup_τ∈Υ|q̃(τ) - q̂(τ)| = o_p(1/√(n)).
 

We first consider the SQR estimator. Note that 

    √(n)(β̂^*(τ) - β(τ)) = _u L^*_n(u,τ),

where L^*_n(u,τ) = ∑_i=1^n [ρ_τ(Y_i^* - Ȧ^*'_iβ(τ) - Ȧ^*'_iu/√(n)) - ρ_τ(Y_i^* - Ȧ^*'_iβ(τ))]. Then, β̂_1^*(τ), the bootstrap counterpart of the SQR estimator, is just the second element of β̂^*(τ). Similar to the proof of Theorem <ref>,

    L^*_n(u,τ) = -u'W_n^*(τ) + Q_n^*(u,τ),

where 

    W_n^*(τ) = ∑_i =1^n 1/√(n)Ȧ_i^*(τ - 1{Y_i^* ≤Ȧ_i^*'β(τ) })

and 

    Q_n^*(u,τ) =    ∑_i=1^n∫_0^Ȧ_i^*' u/√(n)(1{Y_i^* -  Ȧ_i^*'β(τ)≤ v} - 1{Y_i^* -  Ȧ_i^*'β(τ)≤ 0})dv 
    
    =    ∑_i=1^n A^*_i∫_0^u_0+u_1/√(n)(1{Y_i^*(1) - q_1(τ)≤ v} - 1{Y_i^*(1) - q_1(τ)≤ 0})dv 
        + ∑_i=1^n (1-A^*_i)∫_0^u_0/√(n)(1{Y_i^*(0) - q_0(τ)≤ v} - 1{Y_i^*(0) - q_0(τ)≤ 0})dv 
    ≡    Q^*_n,1(u,τ) + Q^*_n,0(u,τ).
 

Define η_i,j^*(s,τ) = (τ - 1{Y_i^*(j)≤ q_j(τ) }) - m_j(s,τ) and η̃_i,j(s,τ) = τ - 1{Y^s_i(j) ≤ q_j(τ)} - m_j(s,τ), j = 0,1, where Y^s_i(j) is defined in the proof of Theorem <ref>. Then, we have 

    W_n^*(τ) =    e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)A^*_i 1{S^*_i = s}(τ - 1{Y^*_i(1) ≤ q_1(τ)})
        + e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (1-A^*_i) 1{S^*_i = s}(τ - 1{Y^*_i(0) ≤ q_0(τ)}) 
    
    =    [e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)A^*_i 1{S^*_i = s}η^*_i,1(s,τ) + e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (1-A^*_i) 1{S^*_i = s}η^*_i,0(s,τ)] 
        + [e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)(A^*_i-π) 1{S^*_i = s}m_1(s,τ) - e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (A_i^*-π) 1{S_i^* = s}m_0(s,τ)] 
        + [e_1 ∑_s ∈𝒮∑_i=1^n 1/√(n)π 1{S^*_i = s}m_1(s,τ) + e_0∑_s ∈𝒮∑_i=1^n 1/√(n) (1-π) 1{S^*_i = s}m_0(s,τ) ] 
    ≡     W^*_n,1(τ) + W^*_n,2(τ) + W^*_n,3(τ).


By Lemma <ref>, there exists a sequence of independent Poisson(1) random variables {ξ_i^s}_i ≥ 1,s∈𝒮 such that {ξ_i^s}_i ≥ 1,s∈𝒮{A_i^*,S_i^*,Y_i,A_i,S_i}_i≥ 1,

    ∑_i =1^nA_i^*1{S_i^*=s}η_i,1^*(s,τ) = ∑_i = N(s)+1^N(s)+n_1(s)ξ_i^sη̃_i,1(s,τ) + R^*_1(s,τ),

and

    ∑_i =1^n(1-A_i^*)1{S_i^*=s}η_i,1^*(s,τ) = ∑_i = N(s)+n_1(s)+1^N(s)+n(s)ξ_i^sη̃_i,0(s,τ) + R^*_0(s,τ),

where sup_τ∈Υ(|R^*_1(s,τ)| + |R^*_0(s,τ)|) = o_p(√(n(s))) = o_p(√(n)) for all s ∈𝒮. Therefore, 

    (W^*_n,1(τ),W^*_n,2(τ),W^*_n,3(τ)) d= (W̃^*_n,1(τ) + R(τ),W^*_n,2(τ), W^*_n,3(τ))

where sup_τ∈Υ||R(τ)|| = o_p(1) and 

    W̃^*_n,1(τ) =  e_1 ∑_s ∈𝒮∑_i = N(s)+1^N(s)+n_1(s)ξ_i^s/√(n)η̃_i,1(s,τ) + e_0∑_s ∈𝒮∑_i = N(s)+n_1(s)+1^N(s)+n(s)ξ_i^s/√(n)η̃_i,0(s,τ)


In addition, following the same argument in the proof of Lemma <ref>, we can further show that 

    W̃^*_n,1(τ) = W^*∗_n,1(τ) + R^∗_n(τ),

where sup_τ∈Υ||R^∗_n(τ)||= o_p(1) and 

    W^*∗_n,1(τ) = e_1 ∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋ξ_i^s/√(n)η̃_i,1(s,τ) + e_0∑_s ∈𝒮∑_i = ⌊ n (F(s)+π p(s)) ⌋+1^⌊ n (F(s)+ p(s)) ⌋ξ_i^s/√(n)η̃_i,0(s,τ).

By construction, W^*∗_n,1(τ)  (W^*_n,2(τ),W^*_n,3(τ)). Also note that {S_i^*}_i=1^n are the nonparametric bootstrap draws based on the empirical CDF of {S_i}_i=1^n. Then, by <cit.>, there exists a sequence of independent Poisson(1) random variables {ξ̃_i}_i ≥ 1 that is independent of data, {A_i^*} and {ξ^s_i}_i ≥ 1,s∈𝒮 such that 

    sup_τ∈Υ||W_n,3^*(τ) - W_n,3^**(τ)|| = o_p(1),

where 

    W_n,3^**(τ) = e_1 ∑_s ∈𝒮∑_i=1^n ξ̃_i/√(n)π 1{S_i = s}m_1(s,τ) + e_0∑_s ∈𝒮∑_i=1^n ξ̃_i/√(n) (1-π) 1{S_i = s}m_0(s,τ)

By Lemma <ref>, 

    Q_n^*(u,τ) 1/2u'Q(τ)u,

where Q(τ) is defined in (<ref>). Then, by the same argument in the proof of Theorem <ref>, we have 

    √(n)(β̂^*(τ) - β(τ)) = Q^-1(τ)(W^*∗_n,1(τ)+W^*_n,2(τ)+W^**_n,3(τ)) + R^*(τ),

where sup_τ∈Υ||R^*(τ)|| = o_p(1). Focusing on the second element of β̂^*(τ), we have 

    √(n)(β̂^*_1(τ) - q(τ)) =    [∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋ξ_i^sη̃_i,1(s,τ)/√(n)π f_1(q_1(τ)) -  ∑_s ∈𝒮∑_i = ⌊ n (F(s)+π p(s)) ⌋+1^⌊ n (F(s)+ p(s)) ⌋ξ_i^sη̃_i,0(s,τ)/√(n)(1-π)f_0(q_0(τ))] 
        + [ ∑_s ∈𝒮D_n^*(s)/√(n)(m_1(s,τ)/π f_1(q_1(τ)) + m_0(s,τ)/π f_0(q_0(τ))) ] 
        + [∑_i=1^n ξ̃_i/√(n)(m_1(S_i,τ) /f_1(q_1(τ)) - m_0(S_i,τ) /f_0(q_0(τ))) ] + R_1^*(τ),

where sup_τ∈Υ|R_1^*(τ)| = o_p(1). In addition, by definition, we have 

    √(n)(q̃(τ) - q(τ)) =     [∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋η̃_i,1(s,τ)/√(n)π f_1(q_1(τ)) -  ∑_s ∈𝒮∑_i = ⌊ n (F(s)+π p(s)) ⌋+1^⌊ n (F(s)+ p(s)) ⌋η̃_i,0(s,τ)/√(n)(1-π)f_0(q_0(τ))] 
        + [∑_i=1^n 1/√(n)(m_1(S_i,τ) /f_1(q_1(τ)) - m_0(S_i,τ) /f_0(q_0(τ))) ].

By taking difference of the two displays above, we have 

    √(n)(β̂^*_1(τ) - q̃(τ)) =    [∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋(ξ_i^s-1)η̃_i,1(s,τ)/√(n)π f_1(q_1(τ)) -  ∑_s ∈𝒮∑_i = ⌊ n (F(s)+π p(s)) ⌋+1^⌊ n (F(s)+ p(s)) ⌋(ξ_i^s-1)η̃_i,0(s,τ)/√(n)(1-π)f_0(q_0(τ))] 
        + [ ∑_s ∈𝒮D_n^*(s)/√(n)(m_1(s,τ)/π f_1(q_1(τ)) + m_0(s,τ)/π f_0(q_0(τ))) ] 
        + [∑_i=1^n ξ̃_i-1/√(n)(m_1(S_i,τ) /f_1(q_1(τ)) - m_0(S_i,τ) /f_0(q_0(τ))) ] + R_1^*(τ).

Note that, conditionally on data, the first and third brackets on the RHS of the above display converge to Gaussian processes with covariance kernels 

    min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)/π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))

and

    𝔼[m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1))] [m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))],

uniformly over τ∈Υ, respectively. In addition, by Assumption <ref>.1, conditionally data (and thus {S_i}_i=1^n), the second bracket on the RHS of (<ref>) converges to a Gaussian process with a covariance kernel 

    𝔼γ(S)[m_1(S,τ_1)m_1(S,τ_2)/π^2 f_1(q_1(τ_1))f_1(q_1(τ_2))+m_1(S,τ_1)m_0(S,τ_2)/π(1-π) f_1(q_1(τ_1))f_0(q_0(τ_2))],

uniformly over τ∈Υ. Furthermore, we notice that these three Gaussian processes are independent. Therefore, we have, conditionally on data and uniformly over τ∈Υ, 

    √(n)(β̂^*_1(τ) - q̃(τ)) ℬ_sqr(τ),

where ℬ_sqr(τ) is defined in Theorem <ref>. This leads to the desired result for the simple quantile regression estimator. 

Next, we briefly describe the derivation for the IPW estimator. Following the proof of Theorem <ref>,  we have 

    √(n)(q̂^*_1(τ) - q_1(τ)) =    _u L^*_n(u,τ),

where 

    L^*_n(u,τ) ≡   ∑_i =1^n A^*_i/π̂^*(S^*_i)[ρ_τ(Y^*_i - q_1(τ) - u/√(n)) - ρ_τ(Y^*_i - q_1(τ))] 
    
    =     - L^*_1,n(τ)u + L^*_2,n(u,τ),

and π̂^*(s) = n_1^*(s)/n^*(s). Then, we have 

    L^*_1,n(τ) = W_ipw,1,n^*(τ) + R_ipw,1^*(τ),

where 

    W_ipw,1,n^*(τ) = ∑_s∈𝒮1/√(n)∑_i=1^n A_i^*1{S_i^*=s}η_i,1^*(s,τ)/π + ∑_i=1^n m_1(S_i^*,τ)/√(n),
 
and 

    R_ipw,1^*(τ) = - ∑_i=1^n ∑_s ∈𝒮A_i^* 1{S_i^*=s}D_n^*(s)/n^*(s)√(n)π̂^*(s)πη_i,1^*(s,τ).

By Lemma <ref>, sup_τ∈Υ|R_ipw,1^*(τ)| = o_p(1). In addition, same as above, we can show that 

    sup_τ∈Υ|W_ipw,1,n^*(τ) - W_ipw,1,n^**(τ)| = o_p(1),

where 

    W_ipw,1,n^**(τ) =     ∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋ξ_i^sη̃_i,1(s,τ)/√(n)π + ∑_i=1^n ξ̃_im_1(S_i,τ) /√(n).

Similar to Lemma <ref>, we can show that, uniformly over τ∈Υ, 

    L_2,n^*(τ) f_1(q_1(τ))u^2/2.

Therefore, 

    √(n)(q̂^*_1(τ) - q_1(τ)) = W_ipw,1,n^**(τ)/f_1(q_1(τ)) + R_ipw,1^**(τ),

where sup_τ∈Υ|R_ipw,1^**(τ)| = o_p(1). Similarly, we can show 

    √(n)(q̂^*_0(τ) - q_0(τ)) = W_ipw,0,n^**(τ)/f_0(q_0(τ)) + R_ipw,0^**(τ),

where sup_τ∈Υ|R_ipw,0^**(τ)| = o_p(1) and 

    W_ipw,0,n^**(τ) =     ∑_s ∈𝒮∑_i = ⌊ n (F(s) π p(s))⌋+1^⌊ n (F(s)+p(s)) ⌋ξ_i^sη̃_i,0(s,τ)/√(n)π + ∑_i=1^n ξ̃_im_0(S_i,τ) /√(n).

Therefore, 

    √(n)(q̂^*(τ) - q̃(τ)) =    [∑_s ∈𝒮∑_i = ⌊ n F(s) ⌋+1^⌊ n (F(s)+π p(s)) ⌋(ξ_i^s-1)η̃_i,1(s,τ)/√(n)π f_1(q_1(τ)) -  ∑_s ∈𝒮∑_i = ⌊ n (F(s)+π p(s)) ⌋+1^⌊ n (F(s)+ p(s)) ⌋(ξ_i^s-1)η̃_i,0(s,τ)/√(n)(1-π)f_0(q_0(τ))] 
        + [∑_i=1^n ξ̃_i-1/√(n)(m_1(S_i,τ) /f_1(q_1(τ)) - m_0(S_i,τ) /f_0(q_0(τ))) ] + R_ipw^*(τ),

where sup_τ∈Υ|R_ipw^*(τ)| = o_p(1). Last, we can show that, conditionally on data and uniformly over τ∈Υ, the RHS of the above display weakly converges to the Gaussian process ℬ_ipw(τ), where ℬ_ipw(τ) is defined in Theorem <ref>. 






§ TECHNICAL LEMMAS



	
	Let S_k be the k-th partial sum of Banach space valued independent identically distributed random variables, then 
	
    ℙ(max_1 ≤ k ≤ n||S_k|| ≥) ≤ 3max_1 ≤ k ≤ nℙ(||S_k||≥/3).


When S_k takes values on , Lemma <ref> is <cit.>. 



	First suppose max_k ℙ(||S_n - S_k||≥ 2/3)≤ 2/3. In addition, define 
	
    A_k = { ||S_k|| ≥, ||S_j||< , 1≤ j < k}.

	
	Then, 
	
    ℙ(max_k ||S_k|| ≥) ≤   ℙ(||S_n|| ≥/3) +  ∑_k=1^nℙ(||S_n|| ≤/3,A_k) 
    ≤   ℙ(||S_n|| ≥/3) +  ∑_k=1^nℙ(||S_n-S_k|| ≥ 2/3)ℙ(A_k) 
    ≤   ℙ(||S_n|| ≥/3) + 2/3ℙ(max_k ||S_k|| ≥).

	This implies, 
	
    ℙ(max_k ||S_k|| ≥) ≤ 3ℙ(||S_n|| ≥/3).

	On the other hand, if max_k ℙ(||S_n - S_k||≥ 2/3) > 2/3, then there exists k_0 such that ℙ(||S_n - S_k_0||≥ 2/3) > 2/3. Thus, 
	
    ℙ(||S_n|| ≥/3) + ℙ(||S_k_0|| ≥/3) ≥ 2/3.

	This implies, 
	
    3max_1 ≤ k ≤ nℙ(||S_k|| ≥/3) ≥ 3 max(ℙ(||S_n|| ≥/3), ℙ(||S_k_0|| ≥/3)) ≥ 1 ≥ℙ(max_1 ≤ k ≤ n||S_k|| ≥).

	This concludes the proof. 



	
	Let W_n,j(τ), j = 1,2,3 be defined as in (<ref>). If Assumptions in Theorem <ref> hold, then uniformly over τ∈Υ, 
	
    (W_n,1(τ), W_n,2(τ), W_n,3(τ))  (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)),

	where (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)) are three independent two-dimensional Gaussian processes with covariance kernels Σ_1(τ_1,τ_2), Σ_2(τ_1,τ_2), and Σ_3(τ_1,τ_2), respectively. The expressions for the three kernels are derived in the proof below. 


	We follow the general argument in the proof of <cit.>. We divide the proof into two steps. In the first step, we show that 
	
    (W_n,1(τ), W_n,2(τ), W_n,3(τ)) d=( W^⋆_n,1(τ), W_n,2(τ), W_n,3(τ)) + o_p(1),

	where the o_p(1) term holds uniformly over τ∈Υ, W^⋆_n,1(τ)  (W_n,2(τ),W_n,3(τ)), and, uniformly over τ∈Υ, 
	
    W^⋆_n,1(τ) ℬ_1(τ).

	In the second step, we show that 
	
    (W_n,2(τ),W_n,3(τ))  (ℬ_2(τ),ℬ_3(τ))

	uniformly over τ∈Υ and ℬ_2(τ) ℬ_3(τ). 
	
	Step 1. Let η̃_i,j(s,τ) = τ - 1{Y_i^s(j) ≤ q_j(τ) } - m_j(s,τ), for j = 0,1, where {Y_i^s(0),Y_i^s(1)}_i ≥ 1 are the same as defined in Step 1 in the proof of Theorem <ref>. In addition, denote 
	
    W̃_n,1(τ) =  e_1 ∑_s ∈𝒮∑_i=N(s) + 1^N(s) + n_1(s)1/√(n)η̃_i,1(s,τ) + e_0∑_s ∈𝒮∑_i=N(s) + n_1(s)+1^N(s) + n(s)1/√(n)η̃_i,0(s,τ).

	Then, we have 
	
    {W_n,1(τ)|{A_i,S_i}_i=1^n}d={W̃_n,1(τ)|{A_i,S_i}_i=1^n}.

	Because both W_n,2(τ) and W_n,3(τ) are only functions of {A_i,S_i}_i=1^n, we have 
	
    (W_n,1(τ),W_n,2(τ),W_n,3(τ)) d= 
    	(W̃_n,1(τ),W_n,2(τ),W_n,3(τ)).

	Let 
	
    W_n,1^⋆(τ) =  e_1 ∑_s ∈𝒮∑_i=⌊ nF(s) ⌋ + 1^⌊ n(F(s)+π p(s)) ⌋1/√(n)η̃_i,1(s,τ) + e_0∑_s ∈𝒮∑_i=⌊ n(F(s)+π p(s)) ⌋+1^⌊ n(F(s)+ p(s)) ⌋1/√(n)η̃_i,0(s,τ).

	Note that W_n,1^⋆(τ) is a function of (Y_i^s(1),Y_i^s(0))_i ≥ 1 only, which is independent of {A_i,S_i}_i=1^n by construction. Therefore, W_n,1^⋆(τ)  (W_n,2(τ),W_n,3(τ)). 
	
	Furthermore, note that 
	
    N(s)/n F(s),   n_1(s)/nπ p(s),   and  n(s)/n p(s).

	Denote Γ_n,j(s,t,τ) = ∑_i =1^⌊ nt ⌋1/√(n)η̃_i,j(s,τ). In order to show sup_τ∈Υ|W̃_n,1(τ) -  W^⋆_n,1(τ)| = o_p(1) and W^⋆_n,1(τ) ℬ_1(τ), it suffices to show that, (1) for j = 0,1 and s ∈𝒮, the stochastic processes 
	
    {Γ_n,j(s,t,τ): t∈ (0,1), τ∈Υ}

	in stochastically equicontinuous; and (2) W^⋆_n,1(τ) converges to ℬ_1(τ) in finite dimension. 
	
	Claim (1). We want to bound
	
    sup|Γ_n,j(s,t_2,τ_2) - Γ_n,j(s,t_1,τ_1)|,

	where supremum is taken over 0 < t_1<t_2<t_1+ < 1 and τ_1<τ_2<τ_1+ such that τ_1,τ_1+∈Υ. Note that,
	
    sup|Γ_n,j(s,t_2,τ_2) - Γ_n,j(s,t_1,τ_1)| 
    ≤   sup_0 < t_1<t_2<t_1+ < 1,τ∈Υ|Γ_n,j(s,t_2,τ) - Γ_n,j(s,t_1,τ)| +  sup_t ∈ (0,1),τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+|Γ_n,j(s,t,τ_2) - Γ_n,j(s,t,τ_1)|.

	Let m = ⌊ nt_2 ⌋ - ⌊ nt_1 ⌋≤⌊ n⌋+1. Then,  for an arbitrary δ>0, by taking = δ^4,  we have
	
    ℙ(sup_0 < t_1<t_2<t_1+ < 1,τ∈Υ|Γ_n,j(s,t_2,τ) - Γ_n,j(s,t_1,τ)| ≥δ) 
    
    	=    ℙ(sup_0 < t_1<t_2<t_1+ < 1,τ∈Υ |∑_i=⌊ nt_1 ⌋+1^i=⌊ nt_2 ⌋η̃_i,j(s,τ)| ≥√(n)δ) 
    
    	=    ℙ(sup_0 < t ≤,τ∈Υ |∑_i=1^⌊ nt ⌋η̃_i,j(s,τ)| ≥√(n)δ) 
    ≤   ℙ(max_1 ≤ k ≤⌊ n⌋sup_τ∈Υ |S_k(τ)| ≥√(n)δ) 
    ≤   270𝔼sup_τ∈Υ |∑_i=1^⌊ n⌋η̃_i,j(s,τ)|/√(n)δ
    ≲   √(n)/√(n)δ≲δ,

	where in the first inequality, S_k(τ) = ∑_i=1^kη̃_i,j(s,τ) and the second inequality holds due to the same argument in (<ref>). For the third inequality, denote 
	
    ℱ = {η̃_i,j(s,τ): τ∈Υ}

	with an envelope function F = 2. In addition, because ℱ is a VC-class with a fixed VC-index, we have
	
    J(1,ℱ) < ∞,

	where 
	
    J(δ,ℱ) = sup_Q ∫_0^δ√(1 + log N(||F||_Q,2,ℱ,L_2(Q)))d,
 
	N(||F||_Q,2,ℱ,L_2(Q)) is the covering number, and the supremum is taken over all discrete probability measures Q. Therefore, by <cit.>
	
    270𝔼sup_τ∈Υ |∑_i=1^⌊ n⌋η̃_i,j(s,τ)|/√(n)δ≲√(⌊ n⌋)[ 𝔼√(⌊ n⌋)||ℙ_⌊ n⌋ - ℙ||_ℱ]/√(n)δ≲√(⌊ n⌋)J(1,ℱ)/√(n)δ.

	For the second term on the RHS of (<ref>), by taking = δ^4, we have 
	
    ℙ(sup_t ∈ (0,1),τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+|Γ_n,j(s,t,τ_2) - Γ_n,j(s,t,τ_1)| ≥δ) 
    
    	=    ℙ(max_1 ≤ k ≤ nsup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+|S_k(τ_1,τ_2)| ≥√(n)δ) 
    ≤   270𝔼sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+ |∑_i=1^n(η̃_i,j(s,τ_2) - η̃_i,j(s,τ_1))|/√(n)δ≲δ√(log(C/δ^2)),

	where in the first equality, S_k(τ_1,τ_2) = ∑_i=1^k (η̃_i,j(s,τ_2) - η̃_i,j(s,τ_1)) and the first inequality follows the same argument as in (<ref>). For the last inequality, denote 
	
    ℱ = {η̃_i,j(s,τ_2) - η̃_i,j(s,τ_1): τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+}

	with a constant envelope function F = C and 
	
    σ^2 = sup_f ∈ℱ𝔼f^2 ∈ [c_1,c_2],

	for some constant 0<c_1<c_2<∞. Last, ℱ is nested by some VC class with a fixed VC index. Therefore, by <cit.>, 
	
    270𝔼sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+ |∑_i=1^n(η̃_i,j(s,τ_2) - η̃_i,j(s,τ_1))|/√(n)δ
    ≲   √(n)𝔼||ℙ_n - ℙ||_ℱ/δ≲√(σ^2 log(C/σ)/δ^2) + Clog(C/σ)/√(n)δ≲δ√(log(C/δ^2)),

	where the last inequality holds by letting n be sufficiently large. Note that δ√(log(C/δ^2))→ 0 as δ→ 0. This concludes the proof of Claim (1). 
	
	Claim (2). For a single τ, by the triangular CLT, 
	
    W_n,1^⋆(τ)  N(0,Σ_1(τ)),

	where Σ_1(τ) = π[τ(1-τ) - 𝔼m^2_1(S,τ)] e_1  e_1^' +  (1-π)[τ(1-τ) -𝔼m^2_0(S,τ)]e_0e_0^'. The convergence in finite dimension can be proved by using the Cramér-Wold device. In particular, we can show that the covariance kernel is 
	
    Σ_1(τ_1,τ_2) =    π[min(τ_1,τ_2) - τ_1τ_2 - 𝔼 m_1(S,τ_1)m_1(S,τ_2)] e_1  e_1^'
        + (1-π)[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)]e_0e_0^'.

	This concludes the proof of Claim (2), and thus leads to the desired results in Step 1.   
	
	Step 2.
	We first consider the marginal distributions for W_n,2(τ) and W_n,3(τ). For W_n,2(τ), by Assumption <ref> and the fact that m_j(s,τ) is continuous in τ∈Υ j=0,1, we have, conditionally on {S_i}_i=1^n, 
	
    W_n,2(τ) =  ∑_s ∈𝒮D_n(s)/√(n)[ e_1 m_1(s,τ) - e_0m_0(s,τ)] ℬ_2(τ),

	where ℬ_2(τ) is a two-dimensional Gaussian process with covariance kernel 
	
    Σ_2(τ_1,τ_2) 
    
    	=    ∑_s ∈𝒮p(s)γ(s)[ e_1 e_1^' m_1(s,τ_1)m_1(s,τ_2) -  e_1 e_0^' m_1(s,τ_1)m_0(s,τ_2) 
       - e_0  e_1^' m_0(s,τ_1)m_1(s,τ_2)+e_0e_0^' m_0(s,τ_1)m_0(s,τ_2) ].

	
	For W_n,3(τ), by the fact that m_j(s,τ) is continuous in τ∈Υ j=0,1, we have that, uniformly over τ∈Υ, 
	
    W_n,3(τ) =  1/√(n)∑_i=1^n[ e_1 π m_1(S_i,τ) + e_0 (1-π)m_0(S_i,τ)] ℬ_3(τ),

	where ℬ_3(τ) a two-dimensional Gaussian process with covariance kernel
	
    Σ_3(τ_1,τ_2) =      e_1  e_1^'π^2 𝔼m_1(S,τ_1)m_1(S,τ_2) +  e_1 e_0^'π(1-π) 𝔼m_1(S,τ_1)m_0(S,τ_2) 
       + e_0  e_1^'π(1-π) 𝔼m_0(S,τ_1)m_1(S,τ_2) + e_0 e_0^' (1-π)^2 𝔼m_0(S,τ_1)m_0(S,τ_2).

	In addition, we note that, for any fixed τ, 
	
    ℙ(W_n,2(τ) ≤ w_1,W_n,3(τ) ≤ w_2) =    𝔼ℙ(W_n,2(τ) ≤ w_1|{S_i}_i=1^n)1{W_n,3(τ) ≤ w_2 }
    
    	=    𝔼ℙ(N(0,Σ_2(τ,τ))≤ w_1)1{W_n,3(τ) ≤ w_2 } + o(1) 
    
    	=    ℙ(N(0,Σ_3(τ,τ))≤ w_2)ℙ(N(0,Σ_2(τ,τ))≤ w_1) + o(1).

	This implies ℬ_2(τ) ℬ_3(τ). By the Cramér-Wold device, we can show that 
	
    (W_n,2(τ),W_n,3(τ))  (ℬ_2(τ),ℬ_3(τ))

	jointly in finite dimension, where by an abuse of notation, ℬ_2(τ) and ℬ_3(τ) have the same marginal distributions of those in (<ref>) and (<ref>), respectively, and ℬ_2(τ) ℬ_3(τ). Last, because both W_n,2(τ) and W_n,3(τ) are tight marginally, so be the joint process (W_n,2(τ),W_n,3(τ)). This concludes the proof of Step 2, and thus the whole lemma. 





	
	Let 𝒲_n,j(τ), j = 1,2 be defined as in (<ref>). If Assumptions in Theorem <ref> hold, then uniformly over τ∈Υ, 
	
    (𝒲_n,1(τ),𝒲_n,2(τ))  (ℬ_ipw,1(τ),ℬ_ipw,2(τ)),

	where (ℬ_ipw,1(τ),ℬ_ipw,2(τ)) are two independent two-dimensional Gaussian processes with covariance kernels Σ_ipw,1(τ_1,τ_2) and Σ_ipw,2(τ_1,τ_2), respectively. The expressions for  Σ_ipw,1(τ_1,τ_2) and Σ_ipw,2(τ_1,τ_2) are derived in the proof below. 


	The proofs of weak convergence and the independence between (ℬ_ipw,1(τ),ℬ_ipw,2(τ)) are similar to that in Lemma <ref>, and thus, are omitted. Next, we focus on deriving the covariance
	kernels. 
	
	First, similar to the argument in the proof of Lemma <ref>, 
	
    𝒲_n,1(τ) d=∑_s ∈𝒮∑_i=N(s)+1^N(s)+n_1(s)1/√(n)f_1(q_1(τ))η̃_i,1(s,τ)-∑_s ∈𝒮∑_i=N(s)+n_1(s)+1^N(s)+n(s)1/√(n)f_0(q_0(τ))η̃_i,0(s,τ).

	Because (η̃_i,1(s,τ),η̃_i,0(s,τ)) are independent across i, n_1(s)/n π p(s), and (n(s) - n_1(s))/n  (1-π)p(s), we have 
	
    Σ_ipw,1(τ_1,τ_2) = min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)  /π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π)f_0(q_0(τ_1))f_0(q_0(τ_2)) .
 
	Obviously, 
	
    Σ_ipw,2(τ_1,τ_2) = 𝔼(m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1)))(m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))),




	
	If Assumptions <ref> and <ref> hold, then conditionally on data,  the second element of [Q(τ)]^-1∑_i=1^n ξ_i-1/√(n)Ȧ_i(τ - 1{Y_i ≤Ȧ'β(τ)}) weakly converges to ℬ̃_sqr(τ), 	where ℬ̃_sqr(τ) is a Gaussian process with covariance kernel Σ̃_sqr(·,·) defined in Theorem <ref>. 


	We denote the second element of 	[Q(τ)]^-1∑_i=1^n ξ_i-1/√(n)Ȧ_i(τ - 1{Y_i ≤Ȧ'β(τ)}) as 
	
    1/√(n)∑_i=1^n (ξ_i - 1)𝒥_i(s,τ),

	where 
	
    𝒥_i(s,τ) = 𝒥_i,1(s,τ) + 𝒥_i,2(s,τ)  + 𝒥_i,3(s,τ),

	
    𝒥_i,1(s,τ) = A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ)),

	
    𝒥_i,2(s,τ) = F_1(s,τ) (A_i - π)1{S_i = s},

	
    F_1(s,τ) =  m_1(s,τ)/π f_1(q_1(τ))+m_0(s,τ)/(1-π) f_0(q_0(τ)),

	and
	
    𝒥_i,3(s,τ) =  (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))1{S_i = s}.

	
	
	
	In order to show the weak convergence, we only need to show (1) conditionally stochastic equicontinuity and (2) conditional convergence in finite dimension. We divide the proof into two steps accordingly. 
	
	Step 1. In order to show the conditionally stochastic equicontinuity, it suffices to show that, for any >0, as n →∞ followed by δ→ 0, 
	
    ℙ_ξ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i(s,τ_2) - 𝒥_i(s,τ_1))| ≥)  0,

	where ℙ_ξ(·) means that the probability operator is with respect to ξ_1,⋯,ξ_n and conditional on data. Note
	
    𝔼ℙ_ξ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i(s,τ_1) - 𝒥_i(s,τ_1))| ≥) 
    
    	=    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i(s,τ_2) - 𝒥_i(s,τ_1))| ≥)
    ≤   ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,1(s,τ_2) - 𝒥_i,1(s,τ_1))| ≥/3) 
        + ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,2(s,τ_2) - 𝒥_i,2(s,τ_1))| ≥/3)
        + ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,3(s,τ_2) - 𝒥_i,3(s,τ_1))| ≥/3).

	
	Further note that 
	
    ∑_i =1^n (ξ_i-1)𝒥_i,1(s,τ) d=∑_i=N(s) + 1^N(s)+n_1(s)(ξ_i-1)η̃_i,1(s,τ)/π f_1(q_1(τ)) - ∑_i=n(s) +n_1(s)+ 1^N(s)+n(s)(ξ_i-1)η̃_i,0(s,τ)/(1-π) f_0(q_0(τ))

	By the same argument in Claim (1) in the proof of Lemma <ref>, we have 
	
    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,1(s,τ_2) - 𝒥_i(s,τ_1))| ≥/3) 
    ≤   3𝔼sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,1(s,τ_2) - 𝒥_i,1(s,τ_1))|/
    ≤   3√(c_2δlog(C/c_1δ)) + 3Clog(C/c_1δ)/√(n)/,
	
	where C, c_1< c_2  are some positive constants that are independent of (n,,δ). By letting n→∞ followed by δ→ 0, the RHS vanishes. 	
	
	For 𝒥_i,2, we note that F_1(s,τ) is Lipschitz in τ. Therefore, 
	
    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,2(s,τ_2) - 𝒥_i,2(s,τ_1))| ≥/3) 
    ≤   ∑_s ∈𝒮ℙ(Cδ|1/√(n)∑_i =1^n (ξ_i - 1)(A_i - π)1{S_i = s}| ≥/3 ) → 0

	as n →∞ followed by δ→ 0, where we use the fact that
	
    sup_s ∈𝒮|1/√(n)∑_i =1^n (ξ_i - 1)(A_i - π)1{S_i = s}| = O_p(1).

	To see this claim, we note that, conditionally on data, 
	
    1/n∑_i=1^n(A_i - π)^21{S_i = s} =    1/n∑_i=1^n(A_i - π - 2π(A_i - π) + π - π^2)1{S_i=s}
    
    	=    D_n(s) - 2π D_n(s)/n + π(1-π)n(s)/nπ (1-π)p(s).

	Then, by the Lindeberg CLT, conditionally on data, 
	
    1/√(n)∑_i =1^n (ξ_i - 1)(A_i - π)1{S_i = s} N(0,π(1-π)p(s)) = O_p(1).

	Last, by the standard maximal inequality (e.g., <cit.>) and the fact that 
	
    (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))

	is Lipschitz in τ, we have, as n →∞ followed by δ→ 0,  
	
    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,3(s,τ_2) - 𝒥_i,3(s,τ_1))| ≥/3) → 0.

	This concludes the proof of the conditionally stochastic equicontinuity. 
	
	
	Step 2.
	We focus on the one-dimension case and aim to show that, conditionally on data, for fixed τ∈Υ,  
	
    1/√(n)∑_s ∈𝒮∑_i =1^n (ξ_i-1) 𝒥_i(s,τ) (0,Σ̃_sqr(τ,τ)).

	The finite-dimensional convergence can be established similarly by the Cramér-Wold device. In view of Lindeberg-Feller central limit theorem, we only need to show that (1)
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 ζ_Y^2(π,τ) + ξ̃_A^2(π,τ) + ξ_S^2(π,τ)

	and (2)
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 𝔼_ξ(ξ-1)^21{|∑_s ∈𝒮(ξ_i - 1)𝒥_i(s,τ)| ≥√(n)}→ 0.

	(2) is obvious as |𝒥_i(s,τ)| is bounded and max_i|ξ_i-1| ≲log(n) as ξ_i is sub-exponential. Next, we focus on (1). We have 
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 
    
    	=    1/n∑_i=1^n∑_s ∈𝒮{[A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] 
        + F_1(s,τ)(A_i -π)1{S_i=s} + [ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))1{S_i = s}]}^2 
    ≡   σ_1^2 + σ_2^2 + σ_3^2 + 2σ_12 + 2σ_13 + 2 σ_23,

	where 
	
    σ_1^2 = 1/n∑_s ∈𝒮∑_i =1^n [A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))]^2,

	
    σ_2^2 = 1/n∑_s ∈𝒮F^2_1(s,τ)∑_i =1^n (A_i-π)^21{S_i = s},

	
    σ_3^2 = 1/n∑_i =1^n [ (m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ)))]^2,

	
    σ_12 = 1/n∑_i=1^n∑_s ∈𝒮[A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))]F_1(s,τ)(A_i - π)1{S_i=s},

	
	
    σ_13 = 1/n∑_i=1^n∑_s ∈𝒮[A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] [ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))],

	and 
	
    σ_23 = σ_12 = 1/n∑_i=1^n∑_s ∈𝒮F_1(s,τ)(A_i - π)1{S_i=s}[ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))].

	For σ_1^2, we have 
	
    σ_1^2 =    1/n∑_s ∈𝒮∑_i =1^n [A_i 1{S_i = s}η^2_i,1(s,τ)/π^2 f^2_1(q_1(τ)) - (1-A_i)1{S_i = s}η^2_i,0(s,τ)/(1-π)^2 f^2_0(q_0(τ))] 
    d=   1/n∑_s ∈𝒮∑_i=N(s)+1^N(s)+n_1(s)η̃^2_i,1(s,τ)/π^2 f^2_1(q_1(τ)) +  1/n∑_s ∈𝒮∑_i=N(s)+n_1(s)+1^N(s)+n(s)η̃^2_i,0(s,τ)/(1-π)^2 f^2_0(q_0(τ))
       τ(1-τ) - 𝔼m_1^s(S,τ)/π f_1^2(q_1(τ)) + τ(1-τ) - 𝔼m_0^s(S,τ)/(1-π) f_0^2(q_0(τ)) = ζ_Y^2(π,τ),

	where the second equality holds due to the rearrangement argument in Lemma <ref> and the convergence in probability holds due to uniform convergence of the partial sum process. 
	
	For σ_2^2, by Assumption <ref>,  
	
    σ_2^2 = 1/n∑_s ∈𝒮F_1^2(s,τ)(D_n(s) - 2π D_n(s) + π(1-π)1{S_i =s}) π(1-π)𝔼F_1^2(S_i,τ) = ξ̃_A^2(π,τ).

	
	For σ_3^2, by the law of large number, 
	
    σ_3^2 𝔼[ (m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ)))]^2 = ξ_S^2(π,τ).

	
	For σ_12, we have 
	
    σ_12 =    1/n∑_s ∈𝒮(1-π)F_1(s,τ)∑_i=1^nA_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - 1/n∑_s ∈𝒮π F_1(s,τ)∑_i=1^n(1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))
    d=   1/n∑_s ∈𝒮(1-π)F_1(s,τ)∑_i=N(s)+1^N(s)+n_1(s)η̃_i,1(s,τ)/π f_1(q_1(τ)) - 1/n∑_s ∈𝒮π F_1(s,τ)∑_i=N(s)+n_1(s)+1^N(s)+n(s)η̃_i,0(s,τ)/(1-π) f_0(q_0(τ)) 0,

	where the last convergence holds because by Lemma <ref>, 
	
    1/n∑_i=N(s)+1^N(s)+n_1(s)η̃_i,1(s,τ)  0,   and  1/n∑_i=N(s)+n_1(s)+1^N(s)+n(s)η̃_i,0(s,τ)  0.

	By the same argument, we can show that 
	
    σ_13 0.

	
	Last, for σ_23, by Assumption <ref>,  
	
    σ_23 = ∑_s ∈𝒮F_1(s,τ)[ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))]D_n(s)/n 0.

	Therefore, conditionally on data, 
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 ζ_Y^2(π,τ) + ξ̃_A^2(π,τ) + ξ_S^2(π,τ).








	
	If Assumptions <ref>.1 and <ref>.2 hold, sup_ s ∈𝒮|D_n^*(s)|/√(n^*(s)) = O_p(1), sup_ s ∈𝒮|D_n(s)|/√(n(s)) = O_p(1), and n(s) →∞ for all s ∈𝒮, a.s., then there exists a sequence of Poisson(1) random variables {ξ_i^s}_i ≥ 1,s∈𝒮 independent of {A_i^*,S_i^*,Y_i,A_i,S_i}_i≥ 1 such that
	
    ∑_i =1^nA_i^*1{S_i^*=s}η_i,1^*(s,τ) = ∑_i = N(s)+1^N(s)+n_1(s)ξ_i^sη̃_i,1(s,τ) + R^*_1(s,τ),

	where sup_τ∈Υ, s ∈𝒮|R^*_1(s,τ)/√(n(s))| = o_p(1). In addition, 
	
    sup_s ∈𝒮,τ∈Υ|∑_i =1^nA_i^*1{S_i^*=s}η_i,1^*(s,τ)|/√(n(s)) = O_p(1).



	Recall {Y^s_i(0),Y^s_i(1)}_i=1^n as defined in the proof of Theorem <ref> and 
	
    η̃_i,j(s,τ) = τ - 1{Y^s_i(j) ≤ q_j(τ)} - m_j(s,τ),
 
	j = 0,1. In addition, let Ψ_n = {η_i,1(s,τ)}_i=1^n, 
	
    ℕ_n = {n(s)/n,n_1(s)/n, n^*(s)/n,n_1^*(s)/n}_s ∈𝒮

	and given ℕ_n, {M_ni}_i=1^n be a sequence of random variables such that the n_1(s) × 1 vector 
	
    M_n^1(s) = (M_n,N(s)+1, ⋯,M_n,N(s)+n_1(s))
 
	and the (n(s) - n_1(s)) × 1 vector 
	
    M_n^0(s) = (M_n,N(s)+n_1(s)+1, ⋯,M_n,N(s)+n(s))
 
	satisfy: 
	
		
  * M_n^1(s) = ∑_i=1^n_1^*(s)m_i and M_n^0(s) = ∑_i=1^n^*(s) - n_1^*(s)m_i', where {m_i}_i=1^n_1^*(s) and {m_i'}_i=1^n^*(s) - n_1^*(s) are n_1^*(s) i.i.d. multinomial(1,n_1^-1(s),⋯,n_1^-1(s)) random vectors and n^*(s) - n_1^*(s) i.i.d. multinomial (1,(n(s) -n_1(s))^-1,⋯,(n(s) -n_1(s))^-1) random vectors, respectively;
		
  * M_n^0(s)  M_n^1(s)|ℕ_n; and 
		
  * {M_n^0(s),M_n^1(s)}_s ∈𝒮 are independent across s given  ℕ_n and are independent of Ψ_n.
	
	
	Recall that, by <cit.>, the original observations can be rearranged according to s ∈𝒮 and then within strata, treatment group first and then the control group. Then, given ℕ_n, Step 3 in Section <ref> implies that the bootstrap observations {Y_i^*}_i=1^n can be generated by drawing with replacement from the empirical distribution of the outcomes in each (s,a) cell for (s,a) ∈𝒮×{0,1}, n^*_a(s) times, a = 0,1, where n_0^*(s) = n^*(s) - n_1^*(s). Therefore,  
	
    ∑_i =1^n A_i^*1{S_i^*=s}η^*_i,1(s,τ) = ∑_i=N(s)+1^N(s)+n_1(s) M_niη̃_i,1(s,τ).

	
	Following the standard approach in dealing with the nonparametric bootstrap, we want to approximate 
    M_ni, i=N(s)+1,⋯,N(s)+n_1(s)
 by a sequence of i.i.d. Poisson(1) random variables. We construct this sequence as follows. Let M_n^1(s) =∑_i=1^N(n_1(s))m_i, where N(k) is a Poisson number with mean k and is independent of ℕ_n. The n_1(s) elements of vector M_n^1(s) is denoted as {M_ni}_i=N(s)+1^N(s)+n_1(s), which is a sequence of i.i.d. Poisson(1) random variables, given ℕ_n. Therefore, 
	
    {M_ni, i=N(s)+1,⋯,N(s)+n_1(s) |ℕ_n}≡{ξ_i^s, i=N(s)+1,⋯,N(s)+n_1(s)|ℕ_n}

	where {ξ^s_i}_i=1^n, s ∈𝒮 are i.i.d. sequences of Poisson(1) random variables such that  {ξ^s_i}_i=1^n are independent across s ∈𝒮 and against ℕ_n. 
	
	Following the argument in <cit.>, given n_1(s), n_1^*(s), and N(n_1(s)) = k, |ξ_i^s - M_ni| is binomially (|k-n_1^*(s)|,n_1(s)^-1)-distributed. In addition, there exists a sequence ℓ_n = O(√(n(s))) such that 
	
    ℙ(|N(n_1(s)) - n_1^*(s)| ≥ℓ_n) ≤   ℙ(|N(n_1(s)) - n_1(s)| ≥ℓ_n/3) + ℙ(|n_1^*(s) - n_1(s)| ≥ 2ℓ_n/3) 
    ≤   𝔼ℙ(|N(n_1(s)) - n_1(s)| ≥ℓ_n/3|n_1(s)) + ℙ(|n_1^*(s) - n_1(s)| ≥ 2ℓ_n/3) 
    ≤   /3 + ℙ(|n_1^*(s) - n_1(s)| ≥ 2ℓ_n/3)  
    ≤   /3 + ℙ(|D_n^*(s)| + |D_n(s)| + π|n^*(s) - n(s)| ≥ 2ℓ_n/3) 
    ≤    2/3 + ℙ(π|n^*(s) - n(s)| ≥ℓ_n/3) 
    ≤   ,
 
	where the first inequality holds due to the union bound inequality, the second inequality holds by the law of iterated expectation, the third inequality holds because (1) conditionally on data, N(n_1(s)) - n_1(s) = O_p(√(n_1(s))) and (2) n_1(s)/n(s) = π+ D_n(s)/n(s)→π>0 as n(s) →∞ , the fourth inequality holds by the fact that 
	
    n_1^*(s) - n_1(s) = D_n^*(s) - D_n(s) + π(n^*(s) - n(s)),

	the fifth inequality holds because by Assumptions <ref> and <ref>,  |D_n^*(s)| + |D_n(s)| = O_p(√(n(s))), and the sixth inequality holds because {S_i^*}_i=1^n is generated from {S_i}_i=1^n by the standard bootstrap procedure, and thus, by <cit.>, 
	
    n^*(s) - n(s) = ∑_i=1^n(M^w_ni-1)(1{S_i = s} - p(s)) = O_p(√(n(s))),

	where (M^w_n1,⋯,M^w_nn) is independent of {S_i}_i=1^n and multinomially distributed with parameters n and (probabilities) 1/n,⋯,1/n. Therefore, by direct calculation, as n →∞,
	
    ℙ(max_N(s)+1 ≤ i ≤ N(s)+n_1(s)|ξ_i^s - M_ni|>2) 
    ≤   ℙ(max_N(s)+1 ≤ i ≤ N(s)+n_1(s)|ξ_i^s - M_ni|>2, n_1(s) ≥ n(s)) + ℙ(n_1(s) ≤ n(s)) 
    ≤    + 𝔼∑_i=N(s)+1^N(s)+n_1(s)ℙ(|ξ_i^s - M_ni|>2,|N(n_1(s)) - n_1^*(s)| ≤ℓ_n, n_1(s) ≥ n(s) |n_1(s),n_1^*(s),n(s)) + 
    ≤    2 +  𝔼n_1(s)ℙ(bin(ℓ_n,n^-1_1(s))>2|n_1(s),n_1^*(s),n(s))1{n_1(s) ≥ n(s) }→ 2,
 
	where we use the fact that 
	
    n_1(s)ℙ(bin(ℓ_n,n^-1_1(s))>2|n_1(s),n_1^*(s),n(s))1{n_1(s) ≥ n(s) }
    ≲    n_1(s)(ℓ_n/n(s))^3(n(s)/n_1(s))^31{n_1(s) ≥ n(s) }≲1/√(n(s))^3→ 0.

	Because  is arbitrary, we have  
	
    ℙ(max_N(s)+1 ≤ i ≤ N(s)+n_1(s)|ξ_i^s - M_ni|>2) → 0.

	
	Note that |ξ_i^s - M_ni| = ∑_j=1^∞ 1{|ξ_i^s - M_ni|≥ j}. Let I_n^j(s) be the set of indexes i ∈{N(s)+1,⋯,N(s)+n_1(s)} such that |ξ_i^s - M_ni|≥ j. Then, ξ_i^s - M_ni = sign(N(n_1(s))-n^*_1(s))∑_j=1^∞ 1{i ∈ I_n^j(s)}. Thus, 
	
    1/√(n(s))∑_i=N(s) + 1^N(s)+n_1(s)(ξ_i^s-M_ni)η̃_i,1(s,τ) = sign(N(n_1(s))-n^*_1(s))∑_j=1^∞[ #I_n^j(s)/√(n(s))1/#I_n^j(s)∑_i ∈ I_n^j(s)η̃_i,1(s,τ)].

	
	In the following, we aim to show that the RHS of (<ref>) converges to zero in probability uniformly over s ∈𝒮,τ∈Υ. First, note that, by (<ref>), max_N(s)+1 ≤ i ≤ N(s)+n_1(s)|ξ_i^s - M_ni| ≤ 2 occurs with probability approaching one. In the event set that max_N(s)+1 ≤ i ≤ N(s)+n_1(s)|ξ_i^s - M_ni| ≤ 2, only the first two terms of the first summation on the RHS of (<ref>) can be nonzero. In addition, for any j, we have j (#I_n^j(s)) ≤ |N(n_1(s)) - n_1(s)| = O_p(√(n(s))), and thus, #I_n^j(s)/√(n(s)) = O_p(1) for j = 1,2. Therefore, it suffices to show that, for j = 1,2,  
	
	
    sup_s ∈𝒮,τ∈Υ|1/#I_n^j(s)∑_i ∈ I_n^j(s)η̃_i,1(s,τ)| = o_p(1).
 
	Note that 
	
    1/#I_n^j(s)∑_i ∈ I_n^j(s)η̃_i,1(s,τ) = ∑_i=N(s) + 1^N(s)+n_1(s)ω_niη̃_i,1(s,τ),

	where ω_ni = 1{|ξ_i^s - M_ni|≥ j}/#I_n^j(s), i=N(s) + 1,⋯,N(s)+n_1(s) and by construction, {ω_ni}_i=N(s) + 1^N(s)+n_1(s) is independent of {η_i,1(s,τ)}_i=1^n. In addition, because {ω_ni}_i=N(s) + 1^N(s)+n_1(s) is exchangeable conditional on ℕ_n, so be it unconditionally. Third, ∑_i=N(s) + 1^N(s)+n_1(s)ω_ni = 1 and max_i=N(s) + 1,⋯,N(s)+n_1(s)|ω_ni| ≤ 1/#I_n^j(s)  0. Then, by the same argument in the proof of <cit.>, for some r ∈ (0,1) and any n_0 = N(s)+1,⋯,N(s)+n_1(s), we have 
	
    𝔼(sup_τ∈Υ, s∈𝒮|∑_i=N(s) + 1^N(s)+n_1(s)ω_niη̃_i,1(s,τ)|^r|Ψ_n,ℕ_n) 
    ≤    (n_0 -1)𝔼[max_N(s)+n_0 ≤ i ≤ N(s)+n_1(s)ω_ni^r|ℕ_n] [1/n_1(s)∑_i=N(s) + 1^N(s)+n_1(s)sup_τ∈Υ, s∈𝒮|η̃_i,1^r(s,τ)|] 
        + (n_1(s)𝔼(ω_ni|ℕ_n))^rmax_n_0 ≤ k ≤ n_1(s)𝔼[sup_τ∈Υ, s∈𝒮|1/k∑_j=N(s)+n_0^N(s)+kη̃_R_j(N(s),n_1(s)),1(s,τ)|^r|ℕ_n,Ψ_n],

	where (R_k_1+1(k_1,k_2),⋯,R_k_1+k_2(k_1,k_2)) is uniformly distributed on the set of all permutations of k_1+1,⋯,k_1+k_2 and independent of ℕ_n and Ψ_n. First note that sup_s ∈𝒮,τ∈Υ|η_i,1(s,τ)| is bounded and 
    max_N(s)+1 ≤ i ≤ N(s)+n_1(s)ω_ni^r ≤ 1/(#I_n^j(s))^r  0.
 Therefore, the first term on the RHS of (<ref>) converges to zero in probability for every fixed n_0. For the second term, because ω_ni|ℕ_n is exchangeable, 
	
    n_1(s)𝔼(ω_ni|ℕ_n) = ∑_i=N(s) + 1^N(s)+n_1(s)𝔼(ω_ni|ℕ_n) = 1.

	
	In addition, let 𝕊_n(k_1,k_2) be the σ-field generated by all functions of {η̃_i,1(s,τ)}_i≥ 1 that are symmetric in their k_1+1 to k_1+k_2 arguments. Then, 
	
    max_n_0 ≤ k ≤ n_1(s)𝔼[sup_τ∈Υ, s∈𝒮|1/k∑_j=N(s)+n_0^N(s)+kη̃_R_j(N(s),n_1(s)),1(s,τ)|^r|ℕ_n, Ψ_n] 
    
    	=    max_n_0 ≤ k ≤ n_1(s)𝔼[sup_τ∈Υ, s∈𝒮|1/k∑_j=N(s)+n_0^N(s)+kη̃_j,1(s,τ)|^r|ℕ_n,𝕊_n(N(s),n_1(s))] 
    ≤    2𝔼{max_n_0 ≤ k [sup_τ∈Υ, s∈𝒮|1/k∑_j=N(s)+1^N(s)+kη̃_j,1(s,τ)|^r]|ℕ_n,𝕊_n(N(s),n_1(s))}
    
    	=     2𝔼{max_n_0 ≤ k [sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)|^r]|ℕ_n,𝕊_n(0,n_1(s))},

	where the inequality holds by the Jansen's inequality and the triangle inequality and the last equality holds because {η̃_j,1(s,τ)}_j≥ 1 is an i.i.d. sequence. Apply expectation on both sides, we obtain that 
	
    𝔼max_n_0 ≤ k ≤ n_1(s)𝔼[sup_τ∈Υ, s∈𝒮|1/k∑_j=N(s)+n_0^N(s)+kη̃_R_j(N(s),n_1(s)),1(s,τ)|^r|ℕ_n, Ψ_n] 
    ≤    2𝔼max_n_0 ≤ k ≤ n[sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)|^r].

	By the usual maximal inequality, as k →∞,
	
    sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)| a.s.⟶ 0,

	which implies that as n_0 →∞
	
    max_n_0 ≤ k ≤ n[sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)|^r] ≤max_n_0 ≤ k [sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)|^r] a.s.⟶ 0.

	In addition, sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)| is bounded. Then, by the bounded convergence theorem, we have, as n_0 →∞, 
	
    𝔼max_n_0 ≤ k ≤ n[sup_τ∈Υ, s∈𝒮|1/k∑_j=1^kη̃_j,1(s,τ)|^r] → 0.

	
	which implies that, 
	
    𝔼max_n_0 ≤ k ≤ n_1(s)𝔼[sup_τ∈Υ, s∈𝒮|1/k∑_j=N(s)+n_0^N(s)+kη̃_R_j(N(s),n_1(s)),1(s,τ)|^r|ℕ_n, Ψ_n]  0.

	Therefore, the second term on the RHS of (<ref>) converges to zero in probability as n_0 →∞. Then, as n →∞ followed by n_0 →∞, 
	
    𝔼(sup_τ∈Υ, s∈𝒮|∑_i=N(s) + 1^N(s)+n_1(s)ω_niη̃_i,1(s,τ)|^r|Ψ_n,ℕ_n)  0.

	Hence, by the Markov inequality and (<ref>), we have 
	
    sup_s ∈𝒮,τ∈Υ|1/#I_n^j(s)∑_i ∈ I_n^j(s)η̃_i,1(s,τ)|  0.

	
	Consequently, following (<ref>)
	
    sup_s ∈𝒮,τ∈Υ|∑_i=N(s) + 1^N(s)+n_1(s)(ξ_i^s-M_ni)η̃_i,1(s,τ)|  = o_p(√(n(s))).

	This concludes the first part of this Lemma. For the second part, we note 
	
    ∑_i=N(s) + 1^N(s)+n_1(s)M_niη̃_i,1(s,τ) d=∑_i=N(s) + 1^N(s)+n_1(s)ξ^s_iη̃_i,1(s,τ) d=∑_i= 1^n_1(s)ξ^s_iη̃_i,1(s,τ),

	where the second equality holds because {ξ_i^s,η̃_i,1(s,τ)}_i ≥ 1{N(s),n_1(s),n(s)}. Then, conditionally on {N(s),n_1(s),n(s)} and uniformly over s ∈𝒮, the usual maximal inequality (<cit.>) implies  
	
    sup_τ∈Υ|∑_i=N(s) + 1^N(s)+n_1(s)M_niη̃_i,1(s,τ)|  d=sup_τ∈Υ|∑_i= 1^n_1(s)ξ^s_iη̃_i,1(s,τ)| = O_p(√(n(s))).

	
	Combining (<ref>), (<ref>), and (<ref>), we establish (<ref>). This concludes the proof. 




	
	If Assumptions <ref>.1 and <ref>.2 hold, sup_ s ∈𝒮|D_n^*(s)|/√(n^*(s)) = O_p(1), sup_ s ∈𝒮|D_n(s)|/√(n(s)) = O_p(1), and n(s) →∞ for all s ∈𝒮, a.s., then, uniformly over τ∈Υ, 
	
    Q^*_n(u,τ) 1/2u'Qu.



	Recall Q^*_n,1(u,τ) and Q^*_n,0(u,τ) defined in (<ref>). We focus on Q^*_n,1(u,τ). Recall the definition of M_ni in the proof of Lemma <ref>. We have  
	
    Q^*_n,1(u,τ) =    ∑_s ∈𝒮∑_i = N(s)+1^N(s)+n_1(s)M_ni∫_0^u_0 + u_1/√(n)(1{Y^s_i(1) - q_1(τ) ≤ v} - 1{Y^s_i(1) - q_1(τ) ≤ 0})dv 
    
    	=    ∑_s ∈𝒮∑_i = N(s)+1^N(s)+n_1(s)M_ni[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s)] + ∑_s ∈𝒮∑_i = N(s)+1^N(s)+n_1(s)M_ni𝔼ϕ_i(u,τ,s),

	where ϕ_i(u,τ,s) = ∫_0^u_0 + u_1/√(n)(1{Y^s_i(1) - q_1(τ) ≤ v} - 1{Y^s_i(1) - q_1(τ) ≤ 0})dv.
	
	Similar to (<ref>), we have
	
    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ] 
    
    	=    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)ξ_i^s[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ] + ∑_s ∈𝒮r_n(u,τ,s),

	where {ξ_i^s}_i=1^n is a sequence of i.i.d. Poisson(1) random variables and is independent of everything else, and 
	
    r_n(u,τ,s) = sign(N(n_1(s)) - n_1^*(s))∑_j=1^∞#I_n^j(s)/√(n(s))1/#I_n^j(s)∑_i ∈ I_n^j(s)√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ].

	We aim to show 
	
    sup_τ∈Υ, s∈𝒮|r_n(u,τ,s)| = o_p(1),

	
	
	Recall that the proof of Lemma <ref> relies on (<ref>) and the fact that 
	
    𝔼sup_n(s) ≥ k ≥ n_0sup_τ∈Υ, s ∈𝒮|1/k∑_j=1^k η̃_j,1(s,τ)| → 0.
 
	Using the same argument and replacing η̃_j,1(s,τ) by √(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ], in order to show (<ref>), we only need to verify that, as n →∞ followed by n_0 →∞, 
	
    𝔼sup_n(s) ≥ k ≥ n_0sup_τ∈Υ, s∈𝒮|1/k∑_i=1^k√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| → 0.

	Note sup_τ∈Υ, s∈𝒮|1/k∑_i=1^k√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| is bounded by |u_0| + |u_1|. It suffices to show that, for any >0, as n(s) →∞ followed by n_0 →∞,  
	
    ℙ(sup_n(s) ≥ k ≥ n_0sup_τ∈Υ, s∈𝒮|1/k∑_i=1^k√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| ≥) → 0.

	Define the class of functions ℱ_n as
	
    ℱ_n = {√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]: τ∈Υ, s∈𝒮}.
 
	Then, ℱ_n is nested by a VC-class with fixed VC-index. In addition, for fixed u, ℱ_n has a bounded (and independent of n) envelope function F = |u_0| + |u_1|. Last, define ℐ_l = {2^l,2^l+1,⋯,2^l+1-1}. Then, 
	
    ℙ(sup_n(s) ≥ k ≥ n_0sup_τ∈Υ, s∈𝒮|1/k∑_i=1^k√(n(s))[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| ≥|n(s)) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n(s)) ⌋+1ℙ(sup_k ∈ℐ_lsup_τ∈Υ, s∈𝒮|1/k∑_i=1^k√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| ≥|n(s)) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n(s)) ⌋+1ℙ(sup_k ≤ 2^l+1sup_τ∈Υ, s∈𝒮|∑_i=1^k√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| ≥ 2^l | n(s)) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n(s)) ⌋+19ℙ(sup_τ∈Υ, s∈𝒮|∑_i=1^2^l+1√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| ≥ 2^l/30 | n(s) ) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n(s)) ⌋+1270 𝔼(sup_τ∈Υ, s∈𝒮|∑_i=1^2^l+1√(n(s))[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ] || n(s))/ 2^l
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n(s)) ⌋+1C_1/ 2^l/2
    ≤   2C_1/√(n_0)→ 0,

	where the first inequality holds by the union bound, the second inequality holds because on ℐ_l, 2^l+1≥ k ≥ 2^l, the third inequality follows the same argument in the proof of Theorem <ref>, the fourth inequality is due to the Markov inequality, the fifth inequality follows the standard maximal inequality such as <cit.> and the constant C_1 is independent of (l,,n), and the last inequality holds by letting n →∞. Because  is arbitrary, we have established (<ref>), and thus, (<ref>), which further implies that 
	
    sup_τ∈Υ, s∈𝒮|r_n(u,τ,s)| = o_p(1).

	In addition, for the leading term of (<ref>), we have 
	
    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)ξ_i^s[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ] 
    
    	=    ∑_s ∈𝒮[ Γ_n^s*(N(s)+n_1(s),τ)- Γ_n^s*(N(s),τ)],

	where 
	
    Γ_n^s*(k,τ,e) =    ∑_i=1^k ξ_i^s∫_0^u_0+u_1/√(n)(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)})dv 
        - k 𝔼[∫_0^u_0+u_1/√(n)(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)})dv].

	By the same argument in (<ref>), we can show that 
	
    sup_0 < t ≤ 1,τ∈Υ|Γ_n^s*(k,τ,e) | = o_p(1),

	where we need to use the fact that the Poisson(1) random variable has an exponential tail and thus
	
    𝔼sup_i∈{1,⋯,n},s∈𝒮ξ_i^s = O(log(n)).

	Therefore, 
	
    sup_τ∈Υ|∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni[ϕ_i(u,τ,s) - 𝔼ϕ_i(u,τ,s) ]| = o_p(1).

	For the second term on the RHS of (<ref>), we have 
	
    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni𝔼ϕ_i(u,τ,s) =    ∑_s ∈𝒮n_1^*(s)𝔼ϕ_i(u,τ,s) 
    
    	=    ∑_s ∈𝒮π p(s) f_1(q_1(τ)|s)/2(u_0+u_1)^2 +o(1) 
    
    	=    π f_1(q_1(τ))(u_0+u_1)^2/2 +o(1),

	where the o(1) term holds uniformly over τ∈Υ, the first equality holds because ∑_i=N(s) + 1^N(s)+n_1(s)M_ni = n^*_1(s) and the second equality holds by the same calculation in (<ref>) and the facts that n^*(s)/n  p(s) and 
	
    n^*_1(s)/n = D_n^*(s)+π n^*(s)/nπ p(s).

	Combining (<ref>)–(<ref>), (<ref>), and (<ref>), we have 
	
    Q^*_n,1(u,τ) π f_1(q_1(τ))(u_0+u_1)^2/2,

	uniformly over τ∈Υ. By the same argument, we can show that, uniformly over τ∈Υ, 
	
    Q^*_n,0(u,τ) (1-π) f_0(q_0(τ))u^2_0/2.

	This concludes the proof. 






	Second Supplement to “Quantile Treatment Effects and Bootstrap Inference under Covariate-Adaptive Randomization": Strata Fixed Effects Quantile Regression Estimation and Additional Simulation Results



This paper gathers the theories for the strata fixed effects quantile regression estimator and additional simulation results. Section <ref> describes the estimation, weighted bootstrap, and covariate-adaptive bootstrap inference procedures for the strata fixed effects quantile regression estimator. Sections <ref>–<ref> prove Theorems <ref>–<ref>, respectively. Section <ref> contains the proofs of the technical lemmas. Section <ref> contains additional simulation results.  

  

 





§ QUANTILE REGRESSION WITH STRATA FIXED EFFECTS


The strata fixed effects estimator for the ATE is obtained by a linear regression of outcome Y_i on the treatment status A_i, controlling for strata dummies {1{S_i=s}_s ∈𝒮}. <cit.> point out that, due to the Frisch-Waugh-Lovell theorem, this estimator is equal to the linear coefficient in the regression of Y_i on Ã_i, in which Ã_i is the residual of the projection of A_i on the strata dummies. Unlike the expectation, the quantile operator is nonlinear. Therefore, we cannot consistently estimate QTEs by a linear QR of Y_i on A_i and strata dummies. Instead, based on the equivalence relationship, we propose to run the QR of Y_i on Ã_i. Formally, let Ã_i = A_i - π̂(S_i) and Ã̇_i = (1,Ã_i)', where π̂(s) = n_1(s)/n(s), n_1(s) = ∑_i=1^n A_i 1{S_i = s}, and n(s) = ∑_i=1^n 1{S_i = s}. Then, the strata fixed effects (SFE) estimator for the QTE is β̂_sfe,1(τ), where   

    β̂_sfe(τ) ≡(β̂_sfe,0(τ),β̂_sfe,1(τ) )'= _b = (b_0,b_1)^'∈^2∑_i =1^n ρ_τ(Y_i - Ã̇_i^' b).
 


	
	If Assumptions <ref>.1–<ref>.3 and <ref> hold and p(s)>0 for s ∈𝒮, then, uniformly over τ∈Υ,  
	
    √(n)(β̂_sfe,1(τ)-q(τ))ℬ_sfe(τ), as n→∞,

	where ℬ_sfe(·) is a Gaussian process with covariance kernel Σ_sfe(·,·). The expression for Σ_sfe(·,·) can be found in the proof of this theorem. 	

In particular, the asymptotic variance for β̂_sfe,1(τ) is 

    ζ_Y^2(π,τ) + ζ_A^' 2(π,τ) + ζ_S^2(τ),

where ζ_Y^2(π,τ) and ζ_S^2(τ) are the same as those defined below Theorem <ref>,

    ζ_A^' 2(π,τ)=    𝔼γ(S)[(m_1(S,τ)-m_0(S,τ))(1-π/π f_1(q_1(τ)) - π/(1-π) f_0(q_0(τ))) 
        + q(τ)(f_1(q_1(τ)|S)/f_1(q_1(τ))-f_0(q_0(τ)|S)/f_0(q_0(τ))) ]^2.



Three remarks are in order. First, if the treatment assignment rule achieves strong balance, then ζ_A^' 2(π,τ) = 0 and the asymptotic variances for  β̂_1(τ) and β̂_sfe,1(τ) are the same. Second, if the treatment assignment rule does not achieve strong balance, then it is difficult to compare the asymptotic variances of β̂_1(τ) and β̂_sfe,1(τ). Based on our simulation results in Section <ref>, the SFE estimator usually has a smaller standard error. Third, in order to analytically compute the asymptotic variance β̂_sfe,1(τ), one needs to nonparametrically estimate not only the unconditional densities f_j(·) but also the conditional densities f_j(·|s) for j = 0,1 and s ∈𝒮. However, such difficulty can be avoided by the covariate-adaptive bootstrap inference considered in Section <ref>.

We can compute the weighted bootstrap counterpart of strata fixed effects estimator: 

    β̂_sfe^w(τ) = _b ∑_i=1^n ξ_i ρ_τ(Y_i - Ã̇_i^w'b),

where Ã̇^w_i = (1,Ã^w_i)', Ã^w_i = A_i - π̂^w(S_i), and π̂^w(·) is defined in Section <ref>. The second element of β̂_sfe^w(τ) is our bootstrap estimator of the QTE. 

	
	If Assumptions <ref>–<ref> hold and p(s)>0 for all s ∈𝒮, then uniformly over τ∈Υ and conditionally on data,
	
    √(n)(β̂_sfe,1^w(τ) - β̂_sfe,1(τ)) ℬ̃_sfe(τ), as n→∞,

	where ℬ̃_sfe(τ) is a Gaussian process with covariance kernel being equal to that of ℬ_sfe(τ) defined in Theorem <ref> with γ(s) being replaced by π(1-π).


Similar to the SQR estimator, the weighted bootstrap fails to capture the cross-sectional dependence due to the covariate-adaptive randomization, and thus, overestimates the asymptotic variance of the SFE estimator. 



We can also implement the covariate-adaptive bootstrap. Let 

    β̂_sfe^*(τ) = _b ∑_i=1^n ρ_τ(Y_i^* - Ã̇_i^*'b),

where Ã̇_i^* = (1,Ã_i^*)', Ã_i^* = A_i^* - π̂^*(S_i^*), π̂^*(s) = n_1^*(s)/n^*(s), and (Y_i^*,A_i^*,S_i^*)_i=1^n is the covariate-adaptive bootstrap sample generated via the procedure mentioned in Section <ref>. The the second element β̂_sfe,1^*(τ) of β̂_sfe^*(τ) is the covariate-adaptive SFE estimator. 


	
	If Assumptions <ref>, <ref>, and <ref> hold and p(s)>0 for all s ∈𝒮, then, uniformly over τ∈Υ and conditionally on data,   
	
    √(n)(β̂_sfe,1^*(τ) - q̂(τ)) ℬ_sfe(τ), as n→∞.

 

Unlike the weighted bootstrap, the covariate-adaptive bootstrap can mimic the cross-sectional dependence, and thus, produces an asymptotically valid standard error for the SFE estimator.  


§ PROOF OF THEOREM <REF>


Define β̃_1(τ) = q(τ), β̃_0(τ) = π q_1(τ) + (1-π)q_0(τ), β̃(τ) = (β̃_0(τ),β̃_1(τ))^', and Ă_i = (1,A_i - π)^'. For arbitrary b_0 and b_1, let u_0 = √(n)(b_0-β̃_0(τ)), u_1 = √(n)(b_1-β̃_1(τ)), u=(u_0, u_1)' ∈^2, and 

    L_sfe,n(u,τ)  = ∑_i=1^n[ρ_τ(Y_i - Ă'_iβ̃(τ) - (Ã̇_i'b  - Ă'_iβ̃(τ))) - ρ_τ(Y_i - Ă'_iβ̃(τ))].
  
Then, by the change of variable, we have that 

    √(n)(β̂_sfe(τ) - β̃(τ)) = _u L_sfe,n(u,τ).

Notice that L_sfe,n(u,τ) is convex in u for each τ and bounded in τ for each u. In the following, we aim to show that there exists 

    g_sfe,n(u,τ) = - u'W_sfe,n(τ) + 1/2u'Q_sfe(τ)u
 
such that (1) for each u, 

    sup_τ∈Υ|L_sfe,n(u,τ) - g_sfe,n(u,τ)-h_sfe,n(τ)|  0,

where h_sfe,n(τ) does not depend on u; 
(2) the maximum eigenvalue of Q_sfe(τ) is bounded from above and the minimum eigenvalue of Q_sfe(τ) is bounded away from 0 uniformly over τ∈Υ; (3) W_sfe,n(τ) ℬ̃(τ) uniformly over τ∈Υ for some ℬ̃(τ).[We abuse the notation and denote the weak limit of W_sfe,n(τ) as ℬ̃(τ). This limit is different from the weak limit of W_n(τ) in the proof of Theorem <ref>.] Then by <cit.>, we have 

    √(n)(β̂_sfe(τ) - β̃(τ)) = [Q_sfe(τ)]^-1W_sfe,n(τ) + r_sfe,n(τ),

where sup_τ∈Υ||r_sfe,n(τ)|| = o_p(1). In addition, by (3), we have, uniformly over τ∈Υ, 

    √(n)(β̂_sfe(τ) - β̃(τ))  [Q_sfe(τ)]^-1ℬ̃(τ) ≡ℬ(τ).

The second element of ℬ(τ) is ℬ_sfe(τ) stated in Theorem <ref>. Next, we prove requirements (1)–(3) in three steps. 

Step 1. By Knight's identity (<cit.>), we have 

    L_sfe,n(u,τ) 
    
    =     -∑_i=1^n(Ã̇_i'(β̃(τ) + u/√(n))  - Ă'_iβ̃(τ))(τ- 1{Y_i≤Ã̇_i'β̃(τ)}) 
        + ∑_i=1^n∫_0^Ã̇_i'(β̃(τ) + u/√(n))  - Ă'_iβ̃(τ)(1{Y_i -  Ã̇_i'β̃(τ)≤ v} - 1{Y_i -  Ã̇_i'β̃(τ)≤ 0})dv
    ≡    -L_1,n(u,τ) + L_2,n(u,τ).


Step 1.1. We first consider L_1,n(u,τ). Note that β̃_1(τ) = q(τ) and 

    L_1,n(u,τ) 
    
    =    ∑_i=1^n∑_s ∈𝒮 A_i 1{S_i = s}(u_0/√(n) + (1-π̂(s))u_1/√(n) + (π - π̂(s))q(τ) )(τ- 1{Y_i(1)≤ q_1(τ)}) 
        +  ∑_i=1^n∑_s ∈𝒮 (1-A_i) 1{S_i = s}(u_0/√(n)  -π̂(s)u_1/√(n) + (π - π̂(s))q(τ) )(τ- 1{Y_i(0)≤ q_0(τ)})  
    ≡     L_1,1,n(u,τ) + L_1,0,n(u,τ).

Let ι_1 = (1,1-π)' and ι_0 = (1,-π)'. Note that π̂(s) - π = D_n(s)/n(s). Then, for L_1,1,n(u,τ), we have 

    L_1,1,n(u,τ) 
    
    =    ∑_i=1^n∑_s ∈𝒮 A_i 1{S_i = s}[u'ι_1/√(n) + (π - π̂(s))(q(τ)+u_1/√(n)) ](τ- 1{Y_i(1)≤ q_1(τ)}) 
    
    =    u'ι_1/√(n)∑_i=1^n∑_s ∈𝒮 A_i 1{S_i = s}(τ- 1{Y_i(1)≤ q_1(τ)}) 
       - ∑_s ∈𝒮D_n(s)/√(n)u_1/n(s)∑_i =1^nA_i1{S_i = s}(τ- 1{Y_i(1)≤ q_1(τ)}) 
        + ∑_s ∈𝒮 (π - π̂(s))q(τ)∑_i =1^nA_i1{S_i = s}(τ- 1{Y_i(1)≤ q_1(τ)}) 
    
    =    ∑_s ∈𝒮u'ι_1/√(n)∑_i=1^n[ A_i 1{S_i = s}η_i,1(s,τ) + (A_i-π) 1{S_i = s}m_1(s,τ) + π 1{S_i = s}m_1(s,τ)] 
        - 
    ∑_s ∈𝒮D_n(s)/√(n)u_1/n(s)∑_i =1^n[A_i 1{S_i=s}η_i,1(s,τ)+ (A_i - π)1{S_i=s}m_1(s,τ) + π1{S_i=s}m_1(s,τ)] + h_1,1(τ) 
    
    =    ∑_s ∈𝒮u'ι_1/√(n)∑_i=1^n[ A_i 1{S_i = s}η_i,1(s,τ) + (A_i-π) 1{S_i = s}m_1(s,τ) + π 1{S_i = s}m_1(s,τ)] 
        - ∑_s ∈𝒮u_1D_n(s) π m_1(s,τ)/√(n) + h_1,1(τ) + R_sfe,1,1(u,τ),

where 

    h_1,1(τ) =  ∑_s ∈𝒮 (π - π̂(s))q(τ)∑_i =1^nA_i1{S_i = s}(τ- 1{Y_i(1)≤ q_1(τ)})

and 

    R_sfe,1,1(u,τ) = - ∑_s ∈𝒮u_1D_n(s)/√(n) n(s)∑_i =1^n[A_i 1{S_i=s}η_i,1(s,τ)+ (A_i - π)1{S_i=s}m_1(s,τ)].

By the same argument in Lemma <ref> and Assumption <ref>.3, we have for every s∈𝒮,

    sup_τ∈Υ|1/√(n)∑_i =1^n A_i 1{S_i=s}η_i,1(s,τ)| = O_p(1)

and 

    sup_τ∈Υ|1/√(n)∑_i =1^n[(A_i - π)1{S_i=s}m_1(s,τ)]| = sup_τ∈Υ|D_n(s)m_1(s,τ)/√(n)| = O_p(1).

In addition, note that n(s)/n  p(s). Therefore, 

    sup_τ∈Υ|R_sfe,1,1(u,τ)| = O_p(1/√(n)) = o_p(1).

Similarly, we have 

    L_1,0,n(u,τ) 
    
    =    ∑_s ∈𝒮u'ι_0/√(n)∑_i=1^n[ (1-A_i) 1{S_i = s}η_i,0(s,τ) - (A_i-π) 1{S_i = s}m_0(s,τ) + (1-π) 1{S_i = s}m_0(s,τ)] 
        - ∑_s ∈𝒮u_1D_n(s) (1-π) m_0(s,τ)/√(n) + h_1,0(τ) + R_sfe,1,0(u,τ),

where 

    h_1,0(τ) =  ∑_s ∈𝒮 (π - π̂(s))q(τ)∑_i =1^n(1-A_i)1{S_i = s}(τ- 1{Y_i(0)≤ q_0(τ)}),


    R_sfe,1,0(u,τ) = - ∑_s ∈𝒮u_1D_n(s)/√(n) n(s)∑_i =1^n[(1-A_i) 1{S_i=s}η_i,0(τ)- (A_i - π)1{S_i=s}m_0(s,τ)],

and 

    sup_τ∈Υ|R_sfe,1,0(τ)| = O_p(1/√(n)) = o_p(1).


Combining (<ref>), (<ref>), (<ref>) and letting ι_2 = (1,1-2π)', we have 

    L_1,n(u,τ) =    1/√(n)∑_s ∈𝒮∑_i =1^n[u'ι_1 A_i 1{S_i=s}η_i,1(s,τ) + u'ι_0(1-A_i)1{S_i=s}η_i,0(s,τ) ] 
        + ∑_s ∈𝒮u'ι_2 D_n(s)/√(n)(m_1(s,τ) - m_0(s,τ)) 
        + 1/√(n)∑_i =1^n(u'ι_1 π m_1(S_i,τ) + u'ι_0 (1-π) m_0(S_i,τ)) 
        + R_sfe,1,1(u,τ)+R_sfe,1,0(u,τ) + h_1,1(τ) + h_1,0(τ).



Step 1.2. Next, we consider L_2,n(u,τ). Denote E_n(s) = √(n)(π̂(s) -π). Then, 

    {E_n(s)}_s ∈𝒮 = {D_n(s)/√(n)n/n(s)}_s ∈𝒮(0,Σ_D') = O_p(1),

where Σ_D'=(γ(s)/p(s):s∈𝒮). In addition,  

    L_2,n(u,τ) 
    
    =    ∑_s ∈𝒮∑_i=1^n A_i 1{S_i = s}∫_0^u'ι_1/√(n) - E_n(s)/√(n)(q(τ)+u_1/√(n))(1{Y_i(1) ≤ q_1(τ) + v} - 1{Y_i(1) ≤ q_1(τ)})dv 
        + ∑_s ∈𝒮∑_i=1^n (1-A_i) 1{S_i = s}∫_0^u'ι_0/√(n) - E_n(s)/√(n)(q(τ)+u_1/√(n))(1{Y_i(0) ≤ q_0(τ) + v} - 1{Y_i(0) ≤ q_0(τ)})dv 
    ≡     L_2,1,n(u,τ) + L_2,0,n(u,τ).

By the same argument in (<ref>), we have 

    L_2,1,n(u,τ) d=   ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)∫_0^u'ι_1/√(n) - E_n(s)/√(n)(q(τ)+u_1/√(n))(1{Y_i^s(1) ≤ q_1(τ) + v} - 1{Y_i^s(1) ≤ q_1(τ)})dv 
    ≡   ∑_s ∈𝒮[Γ_n^s(N(s)+n_1(s),τ,E_n(s)) - Γ_n^s(N(s),τ,E_n(s))],

where 

    Γ_n^s(k,τ,e) = ∑_i=1^k ∫_0^u' ι_1 - e(q(τ)+u_1/√(n))/√(n)(1{Y_i^s(1) ≤ q_1(τ) + v} - 1{Y_i^s(1) ≤ q_1(τ)})dv.

We want to show, for some any sufficiently large constant M, 

    sup_0 < t ≤ 1,τ∈Υ, |e| ≤ M|Γ_n^s(⌊ nt ⌋,τ,e) - 𝔼Γ_n^s(⌊ nt ⌋,τ,e)| = o_p(1).

By the same argument in (<ref>), it suffices to show that 

    sup_τ∈Υ, |e| ≤ Mn||ℙ_n - ℙ||_ℱ = o_p(1),

where 

    ℱ = {∫_0^u' ι_1 - e(q(τ)+u_1/√(n))/√(n)(1{Y_i^s(1) ≤ q_1(τ) + v} - 1{Y_i^s(1) ≤ q_1(τ)})dv: τ∈Υ, |e| ≤ M }

with an envelope F=|u_0| + |u_1| + Msup_τ∈Υ|q(τ)| + |u_1|/√(n)/√(n). Note that  

    sup_f ∈ℱ𝔼f^2 ≤   sup_τ∈Υ𝔼[|u_0| + |u_1| + M|q(τ)| + |u_1|/√(n)/√(n)1{|Y_i^s(1) - q_1(τ)|≤|u_0| + |u_1| + M|q(τ)| + |u_1|/√(n)/√(n)}]^2 
    ≲    n^-3/2,

and ℱ is a VC-class with a fixed VC index. Then, by <cit.>, 

    𝔼sup_τ∈Υ, |e| ≤ M|Γ^s_n(n,τ,e) - 𝔼Γ^s_n(n,τ,e)| = n||ℙ_n  - ℙ||_ℱ≲ n[√(log(n)/n^5/2) + log(n)/n^3/2] = o(1).
 
In addition, we have 

    𝔼Γ_n^s(⌊ nt ⌋,τ,e) =    ⌊ nt ⌋∫_0^u' ι_1 - e(q(τ)+u_1/√(n))/√(n) [F_1(q_1(τ)+v|s) - F_1(q_1(τ)|s)]dv 
    
    =     t f_1(q_1(τ)|s)/2(u' ι_1 - eq(τ))^2 + o(1),

where F_j(·|s) and f_j(·|s), j=0,1 are the conditional CDF and PDF for Y(j) given S=s, respectively, and the o(1) term holds uniformly over {τ∈Υ, |e| ≤ M}. Combining (<ref>) and (<ref>) with the fact that n_1(s)/nπ p(s), we have

    L_2,1,n(u,τ) =    ∑_s ∈𝒮π p(s)f_1(q_1(τ)|s)/2(u' ι_1 - E_n(s)q(τ))^2 + R'_sfe,2,1(u,τ) 
    
    =    π f_1(q_1(τ))/2(u' ι_1)^2 - ∑_s ∈𝒮f_1(q_1(τ)|s)π D_n(s)u' ι_1/√(n)q(τ) + h_2,1(τ) + R_sfe,2,1(u,τ),

where 

    sup_τ∈Υ|R'_sfe,2,1(u,τ)| = o_p(1),   sup_τ∈Υ|R_sfe,2,1(u,τ)| = o_p(1),

and

    h_2,1(τ) =  ∑_s ∈𝒮π f_1(q_1(τ)|s)/2p(s)E^2_n(s)β̃^2_1(τ).


Similarly, we have 

    L_2,0,n(u,τ) =    (1-π)f_0(q_0(τ))/2(u' ι_0)^2 - ∑_s ∈𝒮(1-π)f_0(q_0(τ)|s)D_n(s)u'ι_0/√(n)q(τ) 
         + h_2,0(τ) + R_sfe,2,0(u,τ),

where 

    sup_τ∈Υ|R_sfe,2,0(u,τ)| = o_p(1)   and   h_2,0(τ) =  ∑_s ∈𝒮(1-π)f_0(q_0(τ)|s)/2p(s)E^2_n(s)β̃^2_1(τ).


Combining (<ref>), (<ref>), and (<ref>), we have 

    L_2,n(u,τ) =    1/2u'Q_sfe(τ)u - ∑_s ∈𝒮 q(τ) [f_1(q_1(τ)|s)π u'ι_1 +f_0(q_0(τ)|s)(1-π) u'ι_0 ]D_n(s)/√(n)
        + R_sfe,2,1(u,τ)+R_sfe,2,0(u,τ) + h_2,1(τ) + h_2,0(τ).

where 

    Q_sfe =    π f_1(q_1(τ)) ι_1 ι_1' + (1-π)f_0(q_0(τ))ι_0 ι_0' 
    
    =    [        π f_1(q_1(τ)) + (1-π) f_0(q_0(τ))      π(1-π)(f_1(q_1(τ))  - f_0(q_0(τ)) );      π(1-π)(f_1(q_1(τ))  - f_0(q_0(τ)) ) π(1-π)((1-π)f_1(q_1(τ)) + π f_0(q_0(τ))) ].


Step 1.3. Last, by combining (<ref>) and (<ref>), we have 

    L_sfe,n(u,τ) = - u'W_sfe,n(τ) + 1/2u'Q_sfe(τ)u + R_sfe(u,τ) + h_sfe,n(τ),

where 

    W_sfe,n(τ) 
    
    =    1/√(n)∑_s ∈𝒮∑_i =1^n[ι_1 A_i 1{S_i=s}η_i,1(s,τ) + ι_0(1-A_i)1{S_i=s}η_i,0(s,τ) ] 
        + ∑_s ∈𝒮{ι_2 (m_1(s,τ) - m_0(s,τ)) + q(τ)[f_1(q_1(τ)|s)πι_1 + f_0(q_0(τ)|s)(1-π) ι_0]}D_n(s)/√(n)
        + 1/√(n)∑_i =1^n(ι_1 π m_1(S_i,τ) + ι_0 (1-π) m_0(S_i,τ)) 
    ≡    W_sfe,n,1(τ) + W_sfe,n,2(τ)+W_sfe,n,3(τ),


    R_sfe(u,τ) = R_sfe,1,1(u,τ) + R_sfe,1,0(u,τ) + R_sfe,2,1(u,τ) + R_sfe,2,0(u,τ)

such that sup_τ∈Υ|R_sfe(u,τ)| = o_p(1), and 

    h_sfe,n(τ) = h_1,1(τ)+h_1,0(τ)+h_2,1(τ)+h_2,0(τ).

This concludes the proof of Step 1. 



Step 2. Note that det(Q_sfe(τ)) = π(1-π) f_0(q_0(τ))f_1(q_1(τ)), which is bounded and bounded away from zero. In addition, it can be shown that the two eigenvalues of Q_sfe(τ) are nonnegative. This leads to the desired result. 

Step 3. 
Lemma <ref> establishes the weak convergence that 

    (W_sfe,1,n(τ), W_sfe,2,n(τ), W_sfe,3,n(τ))  (ℬ_sfe,1(τ),ℬ_sfe,2(τ),ℬ_sfe,3(τ)),

where (ℬ_sfe,1(τ),ℬ_sfe,2(τ),ℬ_sfe,3(τ)) are three independent two-dimensional Gaussian processes with covariance kernels Σ_1(τ_1,τ_2), Σ_2(τ_1,τ_2), and Σ_3(τ_1,τ_2), respectively. Therefore, uniformly over τ∈Υ, 

    W_sfe,n(τ) ℬ̃(τ),

where ℬ̃(τ) is a two-dimensional Gaussian process with covariance kernel 

    Σ̃(τ_1,τ_2) = ∑_j=1^3Σ_j(τ_1,τ_2).

Consequently, 

    √(n)(β̂_sfe(τ) - β̃(τ)) ℬ(τ) ≡ Q_sfe^-1(τ)ℬ̃(τ),

where Σ(τ_1,τ_2), the covariance kernel of ℬ(τ), has the expression that 

    Σ(τ_1,τ_2) 
    
    =     Q_sfe^-1(τ_1)Σ̃(τ_1,τ_2)Q_sfe^-1(τ_2) 
    
    =    {1/π f_1(q_1(τ_1))f_1(q_1(τ_2))[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)][ π^2   π;   π   1 ]
        + 1/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)][ (1-π)^2     π-1;     π-1       1 ]}
        + {𝔼γ(S)[(m_1(S,τ_1) - m_0(S,τ_1))[        π/f_0(q_0(τ_1)) + 1-π/f_1(q_1(τ_1)); 1-π/π f_1(q_1(τ_1)) - π/(1-π)f_0(q_0(τ_1)) ] 
    +q(τ_1)f_1(q_1(τ_1)|S)/f_1(q_1(τ_1))[ π; 1 ]
        + q(τ_1)f_0(q_0(τ_1)|S)/f_0(q_0(τ_1))[ 1-π;  -1 ]] ×[(m_1(S,τ_2) - m_0(S,τ_2))[        π/f_0(q_0(τ_2)) + 1-π/f_1(q_1(τ_2)); 1-π/π f_1(q_1(τ_2)) - π/(1-π)f_0(q_0(τ_2)) ]
        +q(τ_2)f_1(q_1(τ_2)|S)/f_1(q_1(τ_2))[ π; 1 ]
    + q(τ_2)f_0(q_0(τ_2)|S)/f_0(q_0(τ_2))[ 1-π;  -1 ]] }
        + {𝔼[m_1(S,τ_1)/f_1(q_1(τ_1))[ π; 1 ] + m_0(S,τ_1)/f_0(q_0(τ_1))[ 1-π;  -1 ]] [m_1(S,τ_2)/f_1(q_1(τ_2))[ π; 1 ] + m_0(S,τ_2)/f_0(q_0(τ_2))[ 1-π;  -1 ]]' }.

By checking the (2,2)-element of Σ(τ_1,τ_2), we have 

    Σ_sfe(τ_1,τ_2) 
    
    =    min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)/π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))
        + 𝔼γ(S)[(m_1(S,τ_1) - m_0(S,τ_1))(1-π/π f_1(q_1(τ_1)) - π/(1-π)f_0(q_0(τ_1))) + q(τ_1)(f_1(q(τ_1)|S)/f_1(q_1(τ_1)) -f_0(q(τ_1)|S)/f_0(q_0(τ_1)))] 
       ×[(m_1(S,τ_2) - m_0(S,τ_2))(1-π/π f_1(q_1(τ_2)) - π/(1-π)f_0(q_0(τ_2))) + q(τ_2)(f_1(q(τ_2)|S)/f_1(q_2(τ_2)) -f_0(q(τ_2)|S)/f_0(q_0(τ_2)))] 
        +𝔼[m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1))] [m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))].




§ PROOF OF THEOREM <REF>


Note that 

    √(n)(β̂_sfe^w(τ)-β̃(τ)) = _u L_sfe,n^w(u,τ),
 
where 

    L_sfe,n^w(u,τ) = ∑_i=1^n ξ_i[ρ_τ(Y_i - Ã̇_i^w '(β̃(τ) + u/√(n))) - ρ_τ(Y_i - Ă_i^'β̃(τ)) ],

Ã̇_i^w = (1,Ã_i^w)', Ã_i^w = A_i - π̂^w(S_i), and 
    π̂^w(s) = ∑_i =1^n ξ_i A_i1{S_i = s}/∑_i =1^n ξ_i 1{S_i = s}.


Similar to the proof of Theorem <ref>, we divide the proof into two steps. In the first step, we show that there exists 

    g_sfe,n^w(u,τ) = - u'W_sfe,n^w(τ) + 1/2u'Q_sfe(τ)u

and h_sfe,n^w(τ) independent of u
such that for each u 

    sup_τ∈Υ|L^w_sfe,n(u,τ) - g_sfe,n^w(u,τ) - h_sfe,n^w(τ)| 0.

In addition, we will show that sup_τ∈Υ||W_sfe,n^w(τ)|| = O_p(1). Then, by <cit.>, we have 

    √(n)(β̂^w_sfe(τ)-β̃(τ)) = [Q_sfe(τ)]^-1W_sfe,n^w(τ)  + R_sfe,n^w(τ),

where 

    sup_τ∈Υ||R_sfe,n^w(τ)|| = o_p(1).

In the second step, we show that, conditionally on data,  

    √(n)(β̂^w_sfe,1(τ)-β̂_sfe,1(τ)) ℬ̃_sfe(τ).


Step 1. Following Step 1 in the proof of Theorem <ref>, we have 

    L^w_sfe,n(u,τ) ≡ - L_1,n^w(u,τ) + L_2,n^w(u,τ),

where 

    L_1,n^w(u,τ) 
    
    =    ∑_i =1^n ∑_s ∈𝒮ξ_iA_i 1{S_i=s}(u_0/√(n) + (1 - π̂^w(s))u_1/√(n) + (π - π̂^w(s))q(τ))(τ - 1{Y_i ≤ q_1(τ)}) 
        + ∑_i =1^n ∑_s ∈𝒮ξ_i(1-A_i)1{S_i=s}(u_0/√(n)-π̂^w(s)u_1/√(n) + (π - π̂^w(s))q(τ))(τ - 1{Y_i ≤ q_0(τ)}) 
    ≡    L_1,1,n^w(u,τ) + L_1,0,n^w(u,τ),


    L_2,n^w(u,τ) 
    
    =    ∑_s ∈𝒮∑_i =1^n ξ_iA_i 1{S_i=s}∫_0^u'ι_1/√(n) - E_n^w(s)/√(n)(q(τ)+u_1/√(n)) (1{Y_i ≤ q_1(τ)+v } - 1{Y_i ≤ q_1(τ)})dv 
        + ∑_s ∈𝒮∑_i =1^n ξ_i(1-A_i) 1{S_i=s}∫_0^u'ι_0/√(n) - E_n^w(s)/√(n)(q(τ)+u_1/√(n)) (1{Y_i ≤ q_0(τ)+v } - 1{Y_i ≤ q_0(τ)})dv 
    ≡    L_2,1,n^w(u,τ) + L_2,0,n^w(u,τ),

and E_n^w(s) = √(n)(π̂^w(s) - π).   

Step 1.1. Recall that ι_1 = (1,1-π)' and ι_0=(1,-π)'. In addition, denote π̂^w(s) - π = D_n^w(s)/n^w(s), where 

    D_n^w(s) = ∑_i =1^n ξ_i(A_i-π)1{S_i = s}  and   n^w(s) = ∑_i =1^n ξ_i1{S_i = s}.

Then, we have 

    L_1,1,n^w(u,τ) 
    
    =    ∑_s ∈𝒮u'ι_1/√(n)∑_i=1^n ξ_i [A_i 1{S_i =s}η_i,1(s,τ)+ π1{S_i=s}m_1(s,τ)] + ∑_s ∈𝒮u'ι_2 D_n^w(s)m_1(s,τ)/√(n)
       +h_1,1^w(τ) + R_sfe,1,1^w(u,τ),

where η_i,1(s,τ) = (τ - 1{Y_i(1) ≤ q_1(τ)}) - m_1(s,τ), 

    h_1,1^w(τ) = ∑_s∈𝒮(π - π̂^w(s))q(τ)(∑_i =1^nξ_i A_i 1{S_i=s}(τ - 1{Y_i ≤ q_1(τ)})),

and 

    R_sfe,1,1^w(u,τ)= - ∑_s ∈𝒮u_1D_n^w(s)/√(n)n^w(s){∑_i =1^n ξ_i[A_i 1{S_i=s}η_i,1(s,τ)+(A_i-π)1{S_i=s}m_1(s,τ)] }.

By Lemma <ref>, we have 

    sup_τ∈Υ|R_sfe,1,1^w(u,τ)| = o_p(1).
 

Similarly, we have 

    L_1,0,n^w(u,τ) 
    
    =    ∑_s ∈𝒮∑_i=1^n ξ_i {u'ι_0/√(n)[(1-A_i) 1{S_i =s}η_i,0(s,τ)+ π1{S_i=s}m_1(s,τ)]  - u'ι_2/√(n)(A_i - π)1{S_i = s}m_0(s,τ)}
       +h_1,0^w(τ) + R_sfe,1,0^w(u,τ),

where 

    sup_τ∈Υ|R_sfe,1,0^w(u,τ)| = o_p(1).

Combining (<ref>) and (<ref>), we have 

    L^w_1,n(u,τ) 
    
    =    1/√(n)∑_s ∈𝒮∑_i =1^n ξ_i[u'ι_1A_i1{S_i = s}η_i,1(u,τ) + u'ι_0(1-A_i)1{S_i = s}η_i,0(u,τ) 
        + u'ι_2 (A_i - π)1{S_i = s}(m_1(s,τ) - m_0(s,τ)) + 1{S_i = s}(u'ι_1 π m_1(s,τ) + u'ι_0(1-π)m_0(s,τ)) ] 
        +R_sfe,1,1^w(u,τ)+R_sfe,1,0^w(u,τ) + h_1,1^w(τ)  +h_1,0^w(τ).


Furthermore, by Lemma <ref>, we have 

    L^w_2,1,n(u,τ) = π f_1(q_1(τ))/2(u'ι_1)^2 - ∑_s ∈𝒮f_1(q_1(τ)|s)π D_n^w(s) u'ι_1/√(n)q(τ) + h_2,1^w(τ) + R_sfe,2,1^w(u,τ)

and 

    L^w_2,0,n(u,τ) = (1-π) f_0(q_0(τ))/2(u'ι_0)^2 - ∑_s ∈𝒮f_0(q_0(τ)|s)(1-π) D_n^w(s) u'ι_0/√(n)q(τ) + h_2,0^w(τ) + R_sfe,2,0^w(u,τ),

where 

    h_2,1^w(τ)  = ∑_s ∈𝒮π f_1(q_1(τ)|s)/2p(s)(E_n^w(s))^2q^2(τ),


    h_2,0^w(τ)  = ∑_s ∈𝒮(1-π) f_0(q_0(τ)|s)/2p(s)(E_n^w(s))^2q^2(τ),


    sup_τ∈Υ|R_sfe,2,1^w(u,τ)| = o_p(1),

and

    sup_τ∈Υ|R_sfe,2,0^w(u,τ)| = o_p(1).

Therefore, 

    L_2,n^w(u,τ) =    1/2u'Q_sfe(τ)u - ∑_s ∈𝒮q(τ)[f_1(q_1(τ)|s)π u'ι_1 + f_0(q_0(τ)|s)(1-π) u'ι_0 ]D_n^w(s)/√(n)
        + R_sfe,2,1^w(u,τ) + R_sfe,2,0^w(u,τ) + h_2,1^w(τ)+h_2,0^w(τ).

Combining (<ref>), (<ref>),  (<ref>), and (<ref>), we have 

    L_sfe,n^w(u,τ) = - u'W̃_sfe,n^w(τ) + 1/2u'Q_sfeu + R̃_sfe,n^w(u,τ) + h_sfe,n^w(τ),

where 

    W_sfe,n^w(τ) 
    
    =    1/√(n)∑_s ∈𝒮∑_i =1^n ξ_i [ι_1A_i 1{S_i=s}η_i,1(s,τ) + ι_0(1-A_i)1{S_i=s}η_i,0(s,τ)] 
        + 1/√(n)∑_s ∈𝒮∑_i =1^n ξ_i{ι_2(m_1(s,τ)- m_0(s,τ)) + q(τ)[f_1(q_1(τ)|s)πι_1 + f_0(q_0(τ)|s)(1-π)ι_0 ]  }
       ×(A_i - π)1{S_i = s} + 1/√(n)∑_i=1^nξ_i(ι_1 π m_1(S_i,τ) + ι_0(1-π)m_0(S_i,τ)),


    h_sfe,n^w(τ) = h_1,1^w(τ)+h_1,0^w(τ)+h_2,1^w(τ)+h_2,0^w(τ),

and

    sup_τ∈Υ|R̃_sfe,n^w(u,τ)| = o_p(1).


In addition, by Lemma <ref>, sup_τ∈Υ|W_sfe,n^w(τ)| = O_p(1). Then, by <cit.>, we have 

    √(n)(β̂^w_sfe(τ)-β̃(τ)) = [Q_sfe(τ)]^-1W_sfe,n^w(τ)  + R_sfe,n^w(τ),

where 

    sup_τ∈Υ||R_sfe,n^w(τ)|| = o_p(1).

This concludes Step 1. 


Step 2. We now focus on the second element of β̂^w_sfe(τ). From Step 1, we know that  

    √(n)(β̂^w_sfe,1(τ) - q(τ)) = 1/√(n)∑_s ∈𝒮∑_i =1^n ξ_i 𝒥_i(s,τ) + 
    R_sfe,n,1^w(τ),

where 

    𝒥_i(s,τ) =    [A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] 
        + {(1-π/π f_1(q_1(τ)) - π/(1-π)f_0(q_0(τ)))(m_1(s,τ)- m_0(s,τ)) 
        + q(τ)[f_1(q_1(τ)|s)/f_1(q_1(τ)) - f_0(q_0(τ)|s)/f_0(q_0(τ))]  }(A_i - π)1{S_i = s}
        + (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))1{S_i = s}

and

    sup_τ∈Υ|R_sfe,n,1^w(τ)| = o_p(1).

By (<ref>), we have  

    √(n)(β̂_sfe,1(τ) - q(τ)) = 1/√(n)∑_s ∈𝒮∑_i =1^n 𝒥_i(s,τ) + R_sfe,n,1(τ),

where 

    sup_τ∈Υ|R_sfe,n,1(τ)|= o_p(1).


Taking the difference of the above two equations, we have 

    √(n)(β̂^w_sfe,1(τ) - β̂_sfe,1(τ)) =  1/√(n)∑_s ∈𝒮∑_i =1^n (ξ_i-1) 𝒥_i(s,τ) + R^w(τ),

where 

    sup_τ∈Υ|R^w(τ)| = o_p(1).

Lemma <ref> shows that, conditionally on data,  

    1/√(n)∑_s ∈𝒮∑_i =1^n (ξ_i-1) 𝒥_i(s,τ) ℬ̃_sfe(τ),

where ℬ̃_sfe(τ) is a Gaussian process with covariance kernel


    Σ̃_sfe(τ_1,τ_2) 
    
    =    min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)/π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))
        + 𝔼π(1-π)[(m_1(S,τ_1) - m_0(S,τ_1))(1-π/π f_1(q_1(τ_1)) - π/(1-π)f_0(q_0(τ_1))) 
        + q(τ_1)(f_1(q(τ_1)|S)/f_1(q_1(τ_1)) -f_0(q(τ_1)|S)/f_0(q_0(τ_1)))] 
       ×[(m_1(S,τ_2) - m_0(S,τ_2))(1-π/π f_1(q_1(τ_2)) - π/(1-π)f_0(q_0(τ_2))) + q(τ_2)(f_1(q(τ_2)|S)/f_1(q_2(τ_2)) -f_0(q(τ_2)|S)/f_0(q_0(τ_2)))] 
        +𝔼[m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1))] [m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))].

This concludes the proof for the SFE estimator. 




§ PROOF OF THEOREM <REF>


Recall the definition of β̃(τ) = (β̃_0(τ),β̃_1(τ))' in the proof of Theorem <ref>. Let u_0 = √(n)(b_0-β̃_0(τ)), u_1 = √(n)(b_1-β̃_1(τ)) and u=(u_0, u_1)' ∈^2. Then, 

    √(n)(β̂^*_sfe(τ)-β̃(τ)) = _u L_sfe,n^*(u,τ),

where 

    L_sfe,n^*(u,τ) = ∑_i =1^n [ ρ_τ(Y_i^* - Ã̇_i^* '(β̃(τ) + u/√(n))) - ρ_τ(Y_i^* - Ă_i^* 'β̃(τ))]

and Ă_i^* = (1, A_i^* - π)'. Following the proof of Theorem <ref>, we divide the current proof into two steps. In the first step, we show that there exist 

    g_sfe,n^*(u,τ) = - u'W_sfe,n^*(τ) + 1/2u'Q_sfe(τ)u

and h_sfe,n^*(τ) independent of u
such that for each u 

    sup_τ∈Υ|L^*_sfe,n(u,τ) - g_sfe,n^*(u,τ) - h_sfe,n^*(τ)| 0.

In addition, we show that sup_τ∈Υ||W_sfe,n^*(τ)|| = O_p(1). Then, by <cit.>, we have 

    √(n)(β̂^*_sfe(τ)-β̃(τ)) = [Q_sfe(τ)]^-1W_sfe,n^*(τ)  + R_sfe,n^*(τ),

where 

    sup_τ∈Υ||R_sfe,n^*(τ)|| = o_p(1).

In the second step, we show that, conditionally on data,  

    √(n)(β̂^*_sfe,1(τ)-q̂(τ)) ℬ_sfe(τ).


Step 1. Following Step 1 in the proof of Theorem <ref>, we have 

    L^*_sfe,n(u,τ) ≡ - L_1,n^*(u,τ) + L_2,n^*(u,τ),

where 

    L_1,n^*(u,τ) 
    
    =    ∑_i =1^n ∑_s ∈𝒮A_i^*1{S_i^*=s}(u_0/√(n) + (1 - π̂^*(s))u_1/√(n) + (π - π̂^*(s))q(τ))(τ - 1{Y_i^* ≤ q_1(τ)}) 
        + ∑_i =1^n ∑_s ∈𝒮(1-A_i^*)1{S_i^*=s}(u_0/√(n)-π̂^*(s)u_1/√(n) + (π - π̂^*(s))q(τ))(τ - 1{Y_i^* ≤ q_0(τ)}) 
    ≡    L_1,1,n^*(u,τ) + L_1,0,n^*(u,τ),


    L_2,n^*(u,τ) 
    
    =    ∑_s ∈𝒮∑_i =1^n A_i^* 1{S_i^*=s}∫_0^u'ι_1/√(n) - E_n^*(s)/√(n)(q(τ)+u_1/√(n)) (1{Y_i^* ≤ q_1(τ)+v } - 1{Y_i^* ≤ q_1(τ)})dv 
        + ∑_s ∈𝒮∑_i =1^n (1-A_i^*) 1{S_i^*=s}∫_0^u'ι_0/√(n) - E_n^*(s)/√(n)(q(τ)+u_1/√(n)) (1{Y_i^* ≤ q_0(τ)+v } - 1{Y_i^* ≤ q_0(τ)})dv 
    ≡    L_2,1,n^*(u,τ) + L_2,0,n^*(u,τ),

and E_n^*(s) = √(n)(π̂^*(s) - π).   

Step 1.1. Recall that ι_1 = (1,1-π)' and ι_0=(1,-π)'. In addition, π̂^*(s) - π = D_n^*(s)/n^*(s). Then, 

    L_1,1,n^*(u,τ) 
    
    =    ∑_s ∈𝒮u'ι_1/√(n)∑_i=1^n [A_i^*1{S_i^*=s}η_i,1^*(s,τ)+ (A_i^* - π)1{S_i^*=s}m_1(s,τ)+ π1{S_i^*=s}m_1(s,τ)]  
        - ∑_s ∈𝒮u_1 D^*_n(s)π m_1(s,τ)/√(n)+h_1,1^*(τ) + R_sfe,1,1^*(u,τ),

where η_i,1^*(s,τ) = (τ - 1{Y_i^*(1) ≤ q_1(τ)}) - m_1(s,τ), 

    h_1,1^*(τ) = ∑_s∈𝒮(π - π̂^*(s))q(τ)(∑_i =1^n A_i^*1{S_i^*=s}(τ - 1{Y_i^* ≤ q_1(τ)})),

and 

    R_sfe,1,1^*(u,τ)= - ∑_s ∈𝒮u_1D_n^*(s)/√(n)n^*(s){∑_i =1^n A_i^*1{S_i^*=s}η_i,1^*(s,τ)+(A_i^*-π)1{S_i^*=s}m_1(s,τ) }.

Note that 

    sup_s ∈𝒮,τ∈Υ|∑_i =1^n(A_i^*-π)1{S_i^*=s}m_1(s,τ)| = sup_s ∈𝒮,τ∈Υ|D_n^*(s)m_1(s,τ)| = O_p(√(n)).

In addition, Lemma <ref> shows   

    sup_s ∈𝒮,τ∈Υ|∑_i =1^nA_i^*1{S_i^*=s}η_i,1^*(s,τ)| = O_p(√(n(s))).

Therefore, we have 

    sup_τ∈Υ|R_sfe,1,1^*(u,τ)| 
    ≤   ∑_s ∈𝒮sup_s ∈𝒮|u_1 D_n^*(s)/√(n)n^*(s)|[sup_s ∈𝒮,τ∈Υ|∑_i =1^nA_i^*1{S_i^*=s}η_i,1^*(s,τ)| +  sup_s ∈𝒮,τ∈Υ|∑_i =1^n(A_i^*-π)1{S_i^*=s}m_1(s,τ)|] 
    
    =     O_p(1/√(n)).


Similarly, we have 

    L_1,0,n^*(u,τ) 
    
    =    ∑_s ∈𝒮u'ι_0/√(n)∑_i=1^n [(1-A_i^*)1{S_i^*=s}η_i,1^*(s,τ) - (A_i^* - π)1{S_i^*=s}m_0(s,τ)+ (1-π)1{S_i^*=s}m_0(s,τ)] 
        - ∑_s ∈𝒮u_1 D^*_n(s)(1-π) m_0(s,τ)/√(n)+h_1,0^*(τ) + R_sfe,1,0^*(u,τ),

where 

    h_1,0^*(τ) = ∑_s∈𝒮(π - π̂^*(s))q(τ)(∑_i =1^n (1-A_i^*)1{S_i^*=s}(τ - 1{Y_i^* ≤ q_0(τ)})),

and 

    R_sfe,1,0^*(u,τ)= - ∑_s ∈𝒮u_1D_n^*(s)/√(n)n^*(s){∑_i =1^n (1-A_i^*)1{S_i^*=s}η_i,0^*(s,τ) -(A_i^*-π)1{S_i^*=s}m_0(s,τ) }

such that 

    sup_τ∈Υ|R_sfe,1,0^*(u,τ)| = O_p(1/√(n)).

Therefore, 

    L_1,n^*(u,τ) =    1/√(n)∑_s ∈𝒮∑_i =1^n[u'ι_1 A_i^*1{S_i^*=s}η_i,1^*(s,τ) + u'ι_0(1-A_i^*)1{S_i^*=s}η_i,0^*(s,τ) ] 
        + ∑_s ∈𝒮 u'ι_2 D_n^*(s)/√(n)(m_1(s,τ) - m_0(s,τ)) 
        + 1/√(n)∑_i=1^n(u'ι_1π m_1(S_i^*,τ) + u'ι_0(1-π)m_0(S_i^*,τ)) 
        + R_sfe,1,1^*(u,τ) + R_sfe,1,0^*(u,τ) + h_1,1(τ) + h_1,0(τ).


Furthermore, by Lemma <ref>, we have 

    L^*_2,1,n(u,τ) = π f_1(q_1(τ))/2(u'ι_1)^2 - ∑_s ∈𝒮f_1(q_1(τ)|s)π D_n^*(s) u'ι_1/√(n)q(τ) + h_2,1^*(τ) + R_sfe,2,1^*(u,τ)

and 

    L^*_2,0,n(u,τ) = (1-π) f_0(q_0(τ))/2(u'ι_0)^2 - ∑_s ∈𝒮f_0(q_0(τ)|s)(1-π) D_n^*(s) u'ι_0/√(n)q(τ) + h_2,0^*(τ) + R_sfe,2,0^*(u,τ),

where 

    h_2,1^*(τ)  = ∑_s ∈𝒮π f_1(q_1(τ)|s)/2p(s)(E_n^*(s))^2q^2(τ),


    h_2,0^*(τ)  = ∑_s ∈𝒮(1-π) f_0(q_0(τ)|s)/2p(s)(E_n^*(s))^2q^2(τ),


    sup_τ∈Υ|R_sfe,2,1^*(u,τ)| = o_p(1),

and

    sup_τ∈Υ|R_sfe,2,0^*(u,τ)| = o_p(1).

Therefore, 

    L_2,n^*(u,τ) =    1/2u'Q_sfe(τ)u - ∑_s ∈𝒮q(τ)[f_1(q_1(τ)|s)π u'ι_1 + f_0(q_0(τ)|s)(1-π) u'ι_0 ]D_n^*(s)/√(n)
        + R_sfe,2,1^*(u,τ) + R_sfe,2,0^*(u,τ) + h_2,1^*(τ)+h_2,0^*(τ).

Combining (<ref>), (<ref>),  (<ref>), and (<ref>), we have 

    L_sfe,n^*(u,τ) = - u'W_sfe,n^*(τ) + 1/2u'Q_sfeu + R̃_sfe,n^*(u,τ) + h_sfe,n^*(τ),

where 

    W_sfe,n^*(τ) 
    
    =    1/√(n)∑_s ∈𝒮∑_i =1^n [ι_1A_i^*1{S_i^*=s}η_i,1^*(s,τ) + ι_0(1-A_i^*)1{S_i^*=s}η_i,0^*(s,τ)] 
        + ∑_s ∈𝒮{ι_2(m_1(s,τ)- m_0(s,τ)) + q(τ)[f_1(q_1(τ)|s)πι_1 + f_0(q_0(τ)|s)(1-π)ι_0 ]  }D_n^*(s)/√(n)
        + 1/√(n)∑_i=1^n(ι_1 π m_1(S_i^*,τ) + ι_0(1-π)m_0(S_i^*,τ)),


    h_sfe,n^*(τ) = h_1,1^*(τ)+h_1,0^*(τ)+h_2,1^*(τ)+h_2,0^*(τ),

and

    sup_τ∈Υ|R̃_sfe,n^*(u,τ)| = o_p(1).


By Lemma <ref>, sup_τ∈Υ|W_sfe,n^*(τ)| = O_p(1). Then, by <cit.>, we have 

    √(n)(β̂^*_sfe(τ)-β̃(τ)) = [Q_sfe(τ)]^-1W_sfe,n^*(τ)  + R_sfe,n^*(τ),

where 

    sup_τ∈Υ||R_sfe,n^*(τ)|| = o_p(1).

This concludes Step 1. 

Step 2. We now focus on the second element of β̂^*_sfe(τ). From Step 1, we know that  

    √(n)(β̂^*_sfe,1(τ) - q(τ)) 
    
    =    1/√(n)∑_s ∈𝒮∑_i =1^n [A_i^*1{S_i^* = s}η_i,1^*(s,τ)/π f_1(q_1(τ)) - (1-A_i^*)1{S_i^* = s}η_i,0^*(s,τ)/(1-π) f_0(q_0(τ))] 
        + ∑_s ∈𝒮{(1-π/π f_1(q_1(τ)) - π/(1-π)f_0(q_0(τ)))(m_1(s,τ)- m_0(s,τ)) + q(τ)[f_1(q_1(τ)|s)/f_1(q_1(τ)) - f_0(q_0(τ)|s)/f_0(q_0(τ))]  }D_n^*(s)/√(n)
        + 1/√(n)∑_i =1^n(m_1(S_i^*,τ)/f_1(q_1(τ)) - m_0(S_i^*,τ)/f_0(q_0(τ))) + R_sfe,n,1^*(τ) 
    ≡    W_sfe,n,1^*(τ) + W_sfe,n,2^*(τ) +W_sfe,n,3^*(τ)  + R_sfe,n,1^*(τ),

where 

    sup_τ∈Υ|R_sfe,n,1^*(τ)| = o_p(1).


By (<ref>), we have  

    √(n)(q̂(τ) - q(τ)) 
    
    =    1/√(n)∑_s ∈𝒮∑_i =1^n [A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] 
        + 1/√(n)∑_i =1^n(m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ))) + R_ipw,n(τ) 
    ≡   𝒲_n,1(τ) + 𝒲_n,2(τ) + R_ipw,n(τ),

where 

    sup_τ∈Υ|R_ipw,n(τ)|= o_p(1).


Taking the difference of the above two equations, we have 

    √(n)(β̂^*_sfe,1(τ) - q̂(τ)) = (W_sfe,n,1^*(τ)-𝒲_n,1(τ)) + W_sfe,n,2^*(τ) +(W_sfe,n,3^*(τ)-𝒲_n,2(τ)) + R^*(τ),

where 

    sup_τ∈Υ|R^*(τ)| = o_p(1).

Lemma <ref> shows that, conditionally on data,  

    (W_sfe,n,1^*(τ)-𝒲_n,1(τ)), W_sfe,n,2^*(τ), (W_sfe,n,3^*(τ)-𝒲_n,2(τ))  (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)),

where (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)) are three independent Gaussian processes and ∑_j=1^3ℬ_j(τ) d=ℬ_sfe(τ). This concludes the proof. 




§ TECHNICAL LEMMAS



	
	Let W_sfe,n,j(τ), j = 1,2,3 be defined as in (<ref>). If Assumptions in Theorem <ref> hold, then uniformly over τ∈Υ, 
	
    (W_sfe,n,1(τ), W_sfe,n,2(τ), W_sfe,n,3(τ))  (ℬ_sfe,1(τ),ℬ_sfe,2(τ),ℬ_sfe,3(τ)),

	where (ℬ_sfe,1(τ),ℬ_sfe,2(τ),ℬ_sfe,3(τ)) are three independent two-dimensional Gaussian process with covariance kernels Σ_sfe,1(τ_1,τ_2), Σ_sfe,2(τ_1,τ_2), and Σ_sfe,3(τ_1,τ_2), respectively. The expressions for the three kernels are derived in the proof below. 


	The proofs of weak convergence and the independence among (ℬ_sfe,1(τ),ℬ_sfe,2(τ),ℬ_sfe,3(τ)) are similar to that in Lemma <ref>, and thus, are omitted. In the following, we focus on deriving the covariance kernels. 
	
	First, similar to the argument in the proof of Lemma <ref>,
	
    W_sfe,n,1(τ) d=ι_1 ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)1/√(n)η̃_i,1(s,τ) + ι_0 ∑_s ∈𝒮∑_i=N(s)+n_1(s) + 1^N(s)+n(s)1/√(n)η̃_i,0(s,τ).

	Therefore, 
	
    Σ_1(τ_1,τ_2) =    π[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)]ι_1ι_1' 
        + (1-π)[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)]ι_0ι_0'.

	
	For W_sfe,n,2(τ), we have 
	
    Σ_2(τ_1,τ_2) =    𝔼γ(S)[ι_2(m_1(S,τ_1) - m_0(S,τ_1))+q(τ_1)(f_1(q_1(τ_1)|S)πι_1 + f_0(q_0(τ_1)|S)(1-π) ι_0) ] 
       ×[ι_2(m_1(S,τ_2) - m_0(S,τ_2))+q(τ_2)(f_1(q_1(τ_2)|S)πι_1 + f_0(q_0(τ_2)|S)(1-π) ι_0) ]'.

	Next, we have 
	
    Σ_3(τ_1,τ_2) = 𝔼(ι_1π m_1(S,τ_1) + ι_0(1-π)m_0(S,τ_1))(ι_1π m_1(S,τ_2) + ι_0(1-π)m_0(S,τ_2))'.

	
	In addition, 
	
    [Q_sfe(τ)]^-1 = [      1-π/f_0(q_0(τ)) + π/f_1(q_1(τ))          1/f_1(q_1(τ))-1/f_0(q_0(τ));          1/f_1(q_1(τ))-1/f_0(q_0(τ)) 1/(1-π)f_0(q_0(τ)) + 1/π f_1(q_1(τ)) ].

	
	Therefore, 
	
    Σ(τ_1,τ_2) 
    
    	=    {1/π f_1(q_1(τ_1))f_1(q_1(τ_2))[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)][ π^2   π;   π   1 ]
        + 1/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2))[min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)][ (1-π)^2     π-1;     π-1       1 ]}
        + {𝔼γ(S)[(m_1(S,τ_1) - m_0(S,τ_1))[        π/f_0(q_0(τ_1)) + 1-π/f_1(q_1(τ_1)); 1-π/π f_1(q_1(τ_1)) - π/(1-π)f_0(q_0(τ_1)) ] 
    	+q(τ_1)f_1(q_1(τ_1)|S)/f_1(q_1(τ_1))[ π; 1 ]
        + q(τ_1)f_0(q_0(τ_1)|S)/f_0(q_0(τ_1))[ 1-π;  -1 ]] ×[(m_1(S,τ_2) - m_0(S,τ_2))[        π/f_0(q_0(τ_2)) + 1-π/f_1(q_1(τ_2)); 1-π/π f_1(q_1(τ_2)) - π/(1-π)f_0(q_0(τ_2)) ]
        +q(τ_2)f_1(q_1(τ_2)|S)/f_1(q_1(τ_2))[ π; 1 ]
    	+ q(τ_2)f_0(q_0(τ_2)|S)/f_0(q_0(τ_2))[ 1-π;  -1 ]] }
        + {𝔼[m_1(S,τ_1)/f_1(q_1(τ_1))[ π; 1 ] + m_0(S,τ_1)/f_0(q_0(τ_1))[ 1-π;  -1 ]] [m_1(S,τ_2)/f_1(q_1(τ_2))[ π; 1 ] + m_0(S,τ_2)/f_0(q_0(τ_2))[ 1-π;  -1 ]]' }.

	



	
	Recall the definition of R^w_sfe,1,1(u,τ) in (<ref>). If Assumptions <ref> and <ref> hold, then 
	
    sup_τ∈Υ|R^w_sfe,1,1(u,τ)| = o_p(1).



	We divide the proof into two steps. In the first step, we show that sup_ s ∈𝒮|D_n^w(s)| = O_p(√(n)). In the second step, we show that 
	
    sup_τ∈Υ,s ∈𝒮|∑_i =1^n ξ_iA_i 1{S_i = s}η_i,1(s,τ)| = O_p(√(n)).

	Then, 
	
    sup_τ∈Υ|R^w_sfe,1,1(u,τ)| 
    ≤   ∑_s ∈𝒮|u_1|/n^w(s)sup_ s ∈𝒮|D_n^w(s)/√(n)|[sup_τ∈Υ, s ∈𝒮|∑_i =1^n ξ_iA_i 1{S_i = s}η_i,1(s,τ)| + sup_ s ∈𝒮|D_n^w(s)| ] 
    
    	=     O_p(1/√(n)),

	as n^w(s)/n  p(s) > 0. 
	
	Step 1. Because 
	
    sup_s ∈𝒮|D_n(s)| = O_p(√(n)),

	we only need to bound the difference D_n^w(s) - D_n(s). Note that 
	
    n(s)^-1/2D_n^w(s)-n(s)^-1/2D_n(s)= n^-1/2∑_i=1^n(ξ_i-1)(A_i-π)1{S_i=s}.

	
	We aim to prove that, if n(s) →∞ and D_n(s)/n(s) = o_p(1), then conditionally on data, for s ∈𝒮, 
	
    n(s)^-1/2∑_i=1^n(ξ_i-1)(A_i-π)1{S_i =s} N(0,π(1-π))

	and they are independent across s∈𝒮. The independence is straightforward because 
	
    1/n(s)∑_i =1^n (ξ_i-1)^2(A_i-π)^21{S_i =s}1{S_i =s'} = 0   for   s ≠ s'.

	
	For the limiting distribution, let 𝒟_n = {Y_i,A_i,S_i}_i=1^n denote data. According to the Lindeberg-Feller central limit theorem, (<ref>) holds because (1)
	
    n(s)^-1∑_i=1^n𝔼[(ξ_i-1)^2(A_i-π)^21{S_i=s}|𝒟_n]=   n(s)^-1∑_i=1^n(A_i-2A_iπ+π^2)1{S_i=s}
    
    	=   n(s)^-1∑_i=1^n(A_i-π-2(A_i-π)π+π-π^2)1{S_i=s}
    
    	=   1-2π/n(s)D_n(s)+π(1-π)
       π(1-π),

	and (2) for every >0, 
	
    n(s)^-1∑_i=1^n(A_i- π)^21{S_i = s}𝔼[(ξ_i-1)^21{|ξ_i-1|(A_i- π)^21{S_i = s}>ε√(n(s))}|𝒟_n]
    ≤   4 𝔼(ξ_i-1)^2 1{2|ξ_i - 1| ≥ε√(n(s))}→ 0,

	where we use the fact that |A_i - π|1{S_i = s}≤ 2 and n(s) →∞. This concludes the proof of Step 1. 
	
	Step 2. By the same rearrangement argument and the fact that {ξ_i}_i=1^n 𝒟_n, we have 
	
    sup_τ∈Υ, s ∈𝒮| 1/n∑_i=1^nξ_i A_i1{S_i = s}η_i,1(s,τ) | d=sup_τ∈Υ, s ∈𝒮| 1/n∑_i=N(s) + 1^N(s)+n_1(s)ξ_i η̃_i,1(s,τ)|.

	Let Γ_n,1(s,t,τ) = ∑_i =1^⌊ nt ⌋ξ_i η̃_i,1(s,τ)/√(n) and ℱ = {ξ_i η̃_i,1(s,τ): τ∈Υ, s ∈𝒮} with envelope F_i = Cξ_i and ||F_i||_P,2<∞. By Lemma <ref> and <cit.>, for any >0, we can choose M sufficiently large such that
	
    ℙ(sup_0 < t ≤ 1,τ∈Υ, s∈𝒮|Γ_n,1(s,t,τ)| ≥ M)≤   270 𝔼sup_τ∈Υ, s∈𝒮|Γ_n,1(s,1,τ)|/M
    
    	=    270 𝔼√(n)||ℙ_n - ℙ||_ℱ/M≲J(1,ℱ)||F_i||_P,2/M < .

	Therefore,   
	
    sup_0 < t ≤ 1,τ∈Υ, s∈𝒮|Γ_n,1(s,t,τ)| = O_p(1)
 
	and 
	
    sup_τ∈Υ, s ∈𝒮| 1/n∑_i=1^nξ_i A_i1{S_i = s}η_i,1(s,τ) | d=   sup_τ∈Υ, s ∈𝒮1/√(n)|Γ_n,1(s,N(s)+n_1(s)/n,τ) - Γ_n,1(s,N(s)/n,τ)| 
    
    	=     O_p(1/√(n)).

	This concludes the proof of Step 2. 




	
	If Assumptions <ref> and <ref> hold, then <ref> and <ref> hold. 


	We focus on (<ref>). Note that 
	
    L_2,1,n^w(u,τ) 
    
    	=    ∑_s ∈𝒮∑_i=1^n ξ_iA_i 1{S_i = s}∫_0^u'ι_1/√(n)- E_n^w(s)/√(n)(q(τ) + u_1/√(n))(1{Y_i(1) ≤ q_1(τ)+v} - 1{Y_i(1) ≤ q_1(τ)})dv 
    
    	=    ∑_s ∈𝒮∑_i=1^nξ_iA_i 1{S_i = s}[ϕ_i(u,τ,s,E_n^w(s)) - 𝔼ϕ_i(u,τ,s,E_n^w(s)|S_i=s)] 
        + ∑_s ∈𝒮∑_i=1^nξ_iA_i 1{S_i=s}𝔼ϕ_i(u,τ,s,E_n^w(s)|S_i=s),
 
	where by Lemma <ref>, E_n^w(s) = √(n)(π̂^w(s) - π) = n/n^w(s)D_n^w(s)/√(n) = O_p(1),
	
    ϕ_i(u,τ,s,e) = ∫_0^u'ι_1/√(n)- e/√(n)(q(τ) + u_1/√(n))(1{Y_i(1) ≤ q_1(τ)+v} - 1{Y_i(1) ≤ q_1(τ)})dv,

	and 𝔼ϕ_i(u,τ,s,E_n^w(s)|S_i=s) is interpreted as 𝔼(ϕ_i(u,τ,s,e)|S_i=s) with e being evaluated at E_n^w(s). 
	
	For the first term on the RHS of (<ref>), by the rearrangement argument in Lemma <ref>, we have
	
    ∑_s ∈𝒮∑_i=1^nξ_iA_i 1{S_i = s}[ϕ_i(u,τ,s,E_n^w(s)) - 𝔼ϕ_i(u,τ,s,E_n^w(s)|S_i=s)]  
    d=   ∑_s ∈𝒮∑_i=N(s)+1^N(s)+n_1(s)ξ_i[ϕ_i^s(u,τ,s,E_n^w(s)) - 𝔼ϕ_i^s(u,τ,s,E_n^w(s))] ,

	where 
	
    ϕ_i^s(u,τ,s,e) = ∫_0^u'ι_1/√(n)- e/√(n)(q(τ) + u_1/√(n))(1{Y^s_i(1) ≤ q_1(τ)+v} - 1{Y^s_i(1) ≤ q_1(τ)})dv.

	Similar to (<ref>), we can show that, as n →∞,   
	
    sup_τ∈Υ, s∈𝒮|∑_i=N(s)+1^N(s)+n_1(s)ξ_i[ϕ_i^s(u,τ,s,E_n^w(s)) - 𝔼ϕ_i^s(u,τ,s,E_n^w(s))]| = o_p(1).

	
	For the second term in (<ref>), we have 
	
    ∑_s ∈𝒮∑_i=1^nξ_i A_i 1{S_i=s}𝔼ϕ_i(u,τ,s,E_n^w(s)|S_i=s) 
    
    	=    ∑_s ∈𝒮∑_i=1^nξ_iπ 1{S_i=s}/n n𝔼ϕ^s_i(u,τ,s,E_n^w(s)) + ∑_s ∈𝒮D_n^w(s)/n n𝔼ϕ^s_i(u,τ,s,E_n^w(s)) 
    
    	=    ∑_s ∈𝒮π p(s) [f_1(q_1(τ)|s)/2(u'ι_1 - E_n^w(s)q(τ))^2 + o_p(1)] + ∑_s ∈𝒮D_n^w(s)/n[f_1(q_1(τ)|s)/2(u'ι_1 - E_n^w(s)q(τ))^2 + o_p(1)] 
    
    	=    π f_1(q_1(τ))/2(u'ι_1)^2 - ∑_s ∈𝒮f_1(q_1(τ)|s)π D_n^w(s)u'ι_1/√(n)q(τ) + h_2,1^w(τ) + o_p(1),

	where the o_p(1) term holds uniformly over (τ,s) ∈Υ×𝒮. The second equality holds by the same calculation in (<ref>) and the fact that ∑_i =1^n ξ_i1{S_i = s}/n  p(s). The last inequality holds because D_n^w(s)/n = o_p(1), E_n^w(s) = n/n^w(s)D_n^w(s)/√(n) = O_p(1), n/n^w(s) 1/p(s), and 
	
    h_2,1^w(τ)  = ∑_s ∈𝒮π f_1(q_1(τ)|s)/2p(s)(E_n^w(s))^2q^2(τ).

	
	Combining (<ref>)–(<ref>), we have 
	
    L_2,1,n^w(u,τ) = π f_1(q_1(τ))/2(u'ι_1)^2 - ∑_s ∈𝒮f_1(q_1(τ)|s)π D_n^w(s)u'ι_1/√(n)q(τ) + h_2,1^w(τ) + R_sfe,2,1^w(u,τ),

	where 
	
    h_2,1^w(τ)  = ∑_s ∈𝒮π f_1(q_1(τ)|s)/2p(s)(E_n^w(s))^2q^2(τ)

	and
	
    sup_τ∈Υ|R_sfe,2,1^w(u,τ)| = o_p(1).

	This concludes the proof. 
	




	
	If Assumptions <ref> and <ref> hold, then sup_τ∈Υ|| W_sfe,n^w(τ) || = O_p(1). 


	It suffices to show that 
	
    sup_τ∈Υ, s ∈𝒮|1/√(n)∑_i =1^n ξ_i A_i 1{S_i=s}η_i,1(s,τ)| = O_p(1)

	
    sup_τ∈Υ, s ∈𝒮|1/√(n)∑_i =1^n ξ_i(1-A_i)1{S_i=s}η_i,0(s,τ)| = O_p(1),

	
    sup_ s ∈𝒮|1/√(n)∑_i =1^nξ_i(A_i - π)1{S_i = s}| =O_p(1),

	and 
	
    sup_τ∈Υ‖1/√(n)∑_i=1^nξ_i(ι_1 π m_1(S_i,τ) + ι_0(1-π)m_0(S_i,τ))‖ = O_p(1).

	Note that (<ref>) holds by the argument in step 2 in the proof of Lemma <ref>, (<ref>) holds similarly, (<ref>) holds by (<ref>) and (<ref>), and (<ref>) holds by the usual maximal inequality, e.g., <cit.>. This concludes the proof. 	



	
	If Assumptions <ref> and <ref> hold, then conditionally on data,  
	
    1/√(n)∑_s ∈𝒮∑_i =1^n (ξ_i-1) 𝒥_i(s,τ) ℬ̃_sfe(τ),

	where ℬ̃_sfe(τ) is a Gaussian process with covariance kernel Σ̃_sfe(·,·) defined in (<ref>). 


	In order to show the weak convergence, we only need to show (1) conditional stochastic equicontinuity and (2) conditional convergence in finite dimension. We divide the proof into two steps accordingly. 
	
	Step 1. In order to show the conditional stochastic equicontinuity, it suffices to show that, for any >0, as n →∞ followed by δ→ 0, 
	
    ℙ_ξ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i(s,τ_2) - 𝒥_i(s,τ_1))| ≥)  0,

	where ℙ_ξ(·) means that the probability operator is with respect to ξ_1,⋯,ξ_n and conditional on data. Note
	
    𝔼ℙ_ξ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i(s,τ_1) - 𝒥_i(s,τ_1))| ≥) 
    
    	=    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i(s,τ_2) - 𝒥_i(s,τ_1))| ≥)
    ≤   ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,1(s,τ_2) - 𝒥_i,1(s,τ_1))| ≥/3) 
        + ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,2(s,τ_2) - 𝒥_i,2(s,τ_1))| ≥/3)
        + ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,3(s,τ_2) - 𝒥_i,3(s,τ_1))| ≥/3),

	where 
	
    𝒥_i,1(s,τ) = A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ)),

	
    𝒥_i,2(s,τ) = F_1(s,τ) (A_i - π)1{S_i = s},

	
    F_1(s,τ) =  (1-π/π f_1(q_1(τ)) - π/(1-π)f_0(q_0(τ)))(m_1(s,τ)- m_0(s,τ)) + q(τ)[f_1(q_1(τ)|s)/f_1(q_1(τ)) - f_0(q_0(τ)|s)/f_0(q_0(τ))],

	
    𝒥_i,3(s,τ) =  (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))1{S_i = s}.

	
	Further note that 
	
    ∑_i =1^n (ξ_i-1)𝒥_i,1(s,τ) d=∑_i=N(s) + 1^N(s)+n_1(s)(ξ_i-1)η̃_i,1(s,τ)/π f_1(q_1(τ)) - ∑_i=N(s) +n_1(s)+ 1^N(s)+n(s)(ξ_i-1)η̃_i,0(s,τ)/(1-π) f_0(q_0(τ))

	By the same argument in Claim (1) in the proof of Lemma <ref>, we have 
	
    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,1(s,τ_2) - 𝒥_i(s,τ_1))| ≥/3) 
    ≤   3𝔼sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,1(s,τ_2) - 𝒥_i,1(s,τ_1))|/
    ≤   3√(c_2δlog(C/c_1δ)) + 3Clog(C/c_1δ)/√(n)/,
	
	where C, c_1< c_2  are some positive constants that are independent of (n,,δ). By letting n→∞ followed by δ→ 0, the RHS vanishes. 	
	
	For 𝒥_i,2, we note that F_1(s,τ) is Lipschitz in τ. Therefore, 
	
    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,2(s,τ_2) - 𝒥_i,2(s,τ_1))| ≥/3) 
    ≤   ∑_s ∈𝒮ℙ(Cδ|1/√(n)∑_i =1^n (ξ_i - 1)(A_i - π)1{S_i = s}| ≥/3 ) → 0

	as n →∞ followed by δ→ 0, in which we use the fact that, by (<ref>), 
	
    sup_s ∈𝒮|1/√(n)∑_i =1^n (ξ_i - 1)(A_i - π)1{S_i = s}| = O_p(1).

	Last, by the standard maximal inequality (e.g., <cit.>) and the fact that 
	
    (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))

	is Lipschitz in τ, we have, as n →∞ followed by δ→ 0,  
	
    ℙ( sup_τ_1,τ_2 ∈Υ, τ_1 < τ_2 < τ_1+δ,s ∈𝒮|1/√(n)∑_i=1^n (ξ_i - 1)(𝒥_i,3(s,τ_2) - 𝒥_i,3(s,τ_1))| ≥/3) → 0

	This concludes the proof of the conditional stochastic equicontinuity. 
	
	
	Step 2.
	We focus on the one-dimension case and aim to show that, conditionally on data, for fixed τ∈Υ,  
	
    1/√(n)∑_s ∈𝒮∑_i =1^n (ξ_i-1) 𝒥_i(s,τ) (0,Σ̃_sfe(τ,τ)).

	The finite-dimensional convergence can be established similarly by the Cramér-Wold device. In view of Lindeberg-Feller central limit theorem, we only need to show that (1)
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 ζ_Y^2(π,τ) + ξ̃_A^' 2(π,τ) + ξ_S^2(π,τ)

	and (2)
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 𝔼_ξ(ξ-1)^21{|∑_s ∈𝒮(ξ_i - 1)𝒥_i(s,τ)| ≥√(n)}→ 0.

	(2) is obvious as |𝒥_i(s,τ)| is bounded and max_i|ξ_i-1| ≲log(n) as ξ_i is sub-exponential. Next, we focus on (1). We have 
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 
    
    	=    1/n∑_i=1^n∑_s ∈𝒮{[A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] 
        + F_1(s,τ)(A_i -π)1{S_i=s} + [ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))1{S_i = s}]}^2 
    ≡   σ_1^2 + σ_2^2 + σ_3^2 + 2σ_12 + 2σ_13 + 2 σ_23,

	where 
	
    σ_1^2 = 1/n∑_s ∈𝒮∑_i =1^n [A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))]^2,

	
    σ_2^2 = 1/n∑_s ∈𝒮F^2_1(s,τ)∑_i =1^n (A_i-π)^21{S_i = s},

	
    σ_3^2 = 1/n∑_i =1^n [ (m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ)))]^2,

	
    σ_12 = 1/n∑_i=1^n∑_s ∈𝒮[A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))]F_1(s,τ)(A_i - π)1{S_i=s},

	
	
    σ_13 = 1/n∑_i=1^n∑_s ∈𝒮[A_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - (1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))] [ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))],

	and 
	
    σ_23 = σ_12 = 1/n∑_i=1^n∑_s ∈𝒮F_1(s,τ)(A_i - π)1{S_i=s}[ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))].

	For σ_1^2, we have 
	
    σ_1^2 =    1/n∑_s ∈𝒮∑_i =1^n [A_i 1{S_i = s}η^2_i,1(s,τ)/π^2 f^2_1(q_1(τ)) - (1-A_i)1{S_i = s}η^2_i,0(s,τ)/(1-π)^2 f^2_0(q_0(τ))] 
    d=   1/n∑_s ∈𝒮∑_i=N(s)+1^N(s)+n_1(s)η̃^2_i,1(s,τ)/π^2 f^2_1(q_1(τ)) +  1/n∑_s ∈𝒮∑_i=N(s)+n_1(s)+1^N(s)+n(s)η̃^2_i,0(s,τ)/(1-π)^2 f^2_0(q_0(τ))
       τ(1-τ) - 𝔼m_1^s(S,τ)/π f_1^2(q_1(τ)) + τ(1-τ) - 𝔼m_0^s(S,τ)/(1-π) f_0^2(q_0(τ)) = ζ_Y^2(π,τ),

	where the second equality holds due to the rearrangement argument in Lemma <ref> and the convergence in probability holds due to uniform convergence of the partial sum process. 
	
	For σ_2^2, by Assumption <ref>,  
	
    σ_2^2 = 1/n∑_s ∈𝒮F_1^2(s,τ)(D_n(s) - 2π D_n(s) + π(1-π)1{S_i =s}) π(1-π)𝔼F_1^2(S_i,τ) = ξ̃_A^' 2(π,τ).

	
	For σ_3^2, by the law of large number, 
	
    σ_3^2 𝔼[ (m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ)))]^2 = ξ_S^2(π,τ).

	
	For σ_12, we have 
	
    σ_12 =    1/n∑_s ∈𝒮(1-π)F_1(s,τ)∑_i=1^nA_i 1{S_i = s}η_i,1(s,τ)/π f_1(q_1(τ)) - 1/n∑_s ∈𝒮π F_1(s,τ)∑_i=1^n(1-A_i)1{S_i = s}η_i,0(s,τ)/(1-π) f_0(q_0(τ))
    d=   1/n∑_s ∈𝒮(1-π)F_1(s,τ)∑_i=N(s)+1^N(s)+n_1(s)η̃_i,1(s,τ)/π f_1(q_1(τ)) - 1/n∑_s ∈𝒮π F_1(s,τ)∑_i=N(s)+n_1(s)+1^N(s)+n(s)η̃_i,0(s,τ)/(1-π) f_0(q_0(τ)) 0,

	where the last convergence holds because by Lemma <ref>, 
	
    1/n∑_i=N(s)+1^N(s)+n_1(s)η̃_i,1(s,τ)  0   and  1/n∑_i=N(s)+n_1(s)+1^N(s)+n(s)η̃_i,0(s,τ)  0.

	By the same argument, we can show that 
	
    σ_13 0.

	
	Last, for σ_23, by Assumption <ref>,  
	
    σ_23 = ∑_s ∈𝒮F_1(s,τ)[ (m_1(s,τ)/f_1(q_1(τ)) - m_0(s,τ)/f_0(q_0(τ)))]D_n(s)/n 0.

	Therefore, we have 
	
    1/n∑_i =1^n[∑_s ∈𝒮𝒥_i(s,τ)]^2 ζ_Y^2(π,τ) + ξ̃_A^' 2(π,τ) + ξ_S^2(π,τ).







	
	Recall R_sfe,2,1^*(u,τ) and R_sfe,2,0^*(u,τ) defined in (<ref>) and (<ref>), respectively. If Assumptions in Theorem <ref> hold, then (<ref>) and (<ref>) hold and 
	
    sup_τ∈Υ|R_sfe,2,1^*(u,τ)| = o_p(1)   and  sup_τ∈Υ|R_sfe,2,0^*(u,τ)| = o_p(1).



	We focus on (<ref>). Following the definition of M_ni in the proof of Lemma <ref> and the argument in the Step 1.2 of the proof of Theorem <ref>, we have 
	
    L_2,1,n^*(u,τ) 
    
    	=    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni∫_0^u'ι_1/√(n)- E_n^*(s)/√(n)(q(τ) + u_1/√(n))(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)})dv 
    
    	=    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni[ϕ_i(u,τ,s,E_n^*(s)) - 𝔼ϕ_i(u,τ,E_n^*(s)) ] + ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni𝔼ϕ_i(u,τ,s,E_n^*(s)),
 
	where E_n^*(s) = √(n)(π̂^*(s) - π) = n/n^*(s)D_n^*(s)/√(n) = O_p(1), 
	
    ϕ_i(u,τ,s,e) = ∫_0^u'ι_1/√(n)- e/√(n)(q(τ) + u_1/√(n))(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)})dv,

	and 𝔼ϕ_i(u,τ,s,E_n^*(s)) is interpreted as 𝔼ϕ_i(u,τ,s,e) with e being evaluated at E_n^*(s). 
	
	For the first term on the RHS of (<ref>), similar to (<ref>), we have
	
    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni[ϕ_i(u,τ,s,E_n^*(s)) - 𝔼ϕ_i(u,τ,s,E_n^*(s)) ] 
    
    	=    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)ξ_i^s[ϕ_i(u,τ,s,E_n^*(s)) - 𝔼ϕ_i(u,τ,s,E_n^*(s)) ] + ∑_s ∈𝒮r_n(u,τ,s,E_n^*(s)),

	where {ξ_i^s}_i=1^n is a sequence of i.i.d. Poisson(1) random variables and is independent of everything else, and 
	
    r_n(u,τ,s,e) = sign(N(n_1(s)) - n_1(s))∑_j=1^∞#I_n^j(s)/√(n)1/#I_n^j(s)∑_i ∈ I_n^j(s)√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ].

	We aim to show 
	
    sup_|e| ≤ M,τ∈Υ, s∈𝒮|r_n(u,τ,s,e)| = o_p(1),

	
	
	Recall that the proof of Lemma <ref> relies on (<ref>) and the fact that 
	
    𝔼sup_n ≥ k ≥ n_0sup_τ∈Υ, s ∈𝒮|1/k∑_j=1^k η̃_j,1(s,τ)| → 0.
 
	Using the same argument and replacing η̃_j,1(s,τ) by √(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ], in order to show (<ref>), we only need to verify that, as n →∞ followed by n_0 →∞, 
	
    𝔼sup_n ≥ k ≥ n_0sup_|e| ≤ M,τ∈Υ, s∈𝒮|1/k∑_j=1^k√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| → 0

	Because sup_|e| ≤ M,τ∈Υ, s∈𝒮|1/k∑_j=1^k√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| is bounded as shown below, it suffices to show that, for any >0, as n →∞ followed by n_0 →∞,  
	
    ℙ(sup_n ≥ k ≥ n_0sup_|e| ≤ M,τ∈Υ, s∈𝒮|1/k∑_j=1^k√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| ≥) → 0.

	Define the class of functions ℱ_n as
	
    ℱ_n = {√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]: |e| ≤ M,τ∈Υ, s∈𝒮}.
 
	Then, ℱ_n is nested by a VC-class with fixed VC-index. In addition, for fixed u, ℱ_n has a bounded (and independent of n) envelope function  
	
    F = |u'ι_1|+M(max_τ∈Υ|q(τ)| + |u_1|).

	Last, define ℐ_l = {2^l,2^l+1,⋯,2^l+1-1}. Then, 
	
    ℙ(sup_n ≥ k ≥ n_0sup_|e| ≤ M,τ∈Υ, s∈𝒮|1/k∑_j=1^k√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| ≥) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n) ⌋+1ℙ(sup_k ∈ℐ_lsup_|e| ≤ M,τ∈Υ, s∈𝒮|1/k∑_j=1^k√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| ≥) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n) ⌋+1ℙ(sup_k ≤ 2^l+1sup_|e| ≤ M,τ∈Υ, s∈𝒮|∑_j=1^k√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| ≥ 2^l ) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n) ⌋+19ℙ(sup_|e| ≤ M,τ∈Υ, s∈𝒮|∑_j=1^2^l+1√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]| ≥ 2^l/30 ) 
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n) ⌋+1270 𝔼sup_|e| ≤ M,τ∈Υ, s∈𝒮|∑_j=1^2^l+1√(n)[ϕ_i(u,τ,s,e) - 𝔼ϕ_i(u,τ,s,e) ]|/ 2^l
    ≤   ∑_l=⌊log_2(n_0) ⌋^⌊log_2(n) ⌋+1C_1/ 2^l/2
    ≤   2C_1/√(n_0)→ 0,

	where the first inequality holds by the union bound, the second inequality holds because on ℐ_l, 2^l+1≥ k ≥ 2^l, the third inequality follows the same argument in the proof of Theorem <ref>, the fourth inequality is due to the Markov inequality, the fifth inequality follows the standard maximal inequality such as <cit.> and the constant C_1 is independent of (l,,n), and the last inequality holds by letting n →∞. Because  is arbitrary, we have established (<ref>), and thus, (<ref>), which further implies that 
	
    sup_τ∈Υ, s∈𝒮|r_n(u,τ,s,E_n^*(s))| = o_p(1),

	For the leading term of (<ref>), we have 
	
    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)ξ_i^s[ϕ_i(u,τ,s,E_n^*(s)) - 𝔼ϕ_i(u,τ,s,E_n^*(s)) ] 
    
    	=    ∑_s ∈𝒮[ Γ_n^s*(N(s),τ,E_n^*(s)) - Γ_n^s*(N(s)+n_1(s),τ,E_n^*(s))],

	where 
	
    Γ_n^s*(k,τ,e) =    ∑_i=1^k ξ_i^s∫_0^u'ι_1 - e(q(τ)+u_1/√(n))/√(n)(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)})dv 
        - k 𝔼[∫_0^u'ι_1 - e(q(τ)+u_1/√(n))/√(n)(1{Y_i^s(1) ≤ q_1(τ)+v} - 1{Y_i^s(1) ≤ q_1(τ)})dv].

	By the same argument in (<ref>), we can show that 
	
    sup_0 < t ≤ 1,τ∈Υ, |e| ≤ M|Γ_n^s*(k,τ,e) | = o_p(1),

	where we need to use the fact that the Poisson(1) random variable has an exponential tail and thus
	
    𝔼sup_i∈{1,⋯,n},s∈𝒮ξ_i^s = O(log(n)).

	Therefore, 
	
    sup_τ∈Υ|∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni[ϕ_i(u,τ,s,E_n^*(s)) - 𝔼ϕ_i(u,τ,E_n^*(s)) ]| = o_p(1).

	For the second term on the RHS of (<ref>), we have 
	
    ∑_s ∈𝒮∑_i=N(s) + 1^N(s)+n_1(s)M_ni𝔼ϕ_i(u,τ,s,e) =    ∑_s ∈𝒮n_1^*(s)𝔼ϕ_i(u,τ,s,e) 
    
    	=    ∑_s ∈𝒮π p(s) f_1(q_1(τ)|s)/2(u'ι_1 - eq(τ))^2 +o(1),

	where the o(1) term holds uniformly over (τ,e) ∈Υ× [-M,M], the first equality holds because ∑_i=N(s) + 1^N(s)+n_1(s)M_ni = n^*_1(s) and the second equality holds by the same calculation in (<ref>) and the facts that n^*(s)/n  p(s) and 
	
    n^*_1(s)/n = D_n^*(s)+π n^*(s)/nπ p(s).

	
	Combining (<ref>), (<ref>), (<ref>), (<ref>), and the facts that E_n^*(s) = n/n^*(s)D_n^*(s)/√(n) and n/n^*(s) 1/p(s), we have 
	
    L_2,1,n^*(u,τ) = π f_1(q_1(τ))/2(u'ι_1)^2 - ∑_s ∈𝒮f_1(q_1(τ)|s)π D_n^*(s)u'ι_1/√(n)q(τ) + h_2,1^*(τ) + R_sfe,2,1^*(u,τ),

	where 
	
    h_2,1^*(τ)  = ∑_s ∈𝒮π f_1(q_1(τ)|s)/2p(s)(E_n^*(s))^2q^2(τ)

	and
	
    sup_τ∈Υ|R_sfe,2,1^*(u,τ)| = o_p(1).

	This concludes the proof. 



	
	Recall the definition of (W_sfe,n,1^*(τ)-𝒲_n,1(τ), W_sfe,n,2^*(τ), W_sfe,n,3^*(τ)-𝒲_n,2(τ)) in (<ref>). If Assumptions in Theorem <ref> hold, then conditionally on data, 
	
    (W_sfe,n,1^*(τ)-𝒲_n,1(τ), W_sfe,n,2^*(τ), W_sfe,n,3^*(τ)-𝒲_n,2(τ))  (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)),

	where (ℬ_1(τ),ℬ_2(τ),ℬ_3(τ)) are three independent Gaussian processes with covariance kernels
	
    Σ_1(τ_1,τ_2) = min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_1(S,τ_1)m_1(S,τ_2)/π f_1(q_1(τ_1))f_1(q_1(τ_2)) + min(τ_1,τ_2) - τ_1τ_2 - 𝔼m_0(S,τ_1)m_0(S,τ_2)/(1-π) f_0(q_0(τ_1))f_0(q_0(τ_2)),

	
    Σ_2(τ_1,τ_2) 
    
    	=    𝔼γ(S)[(m_1(S,τ_1) - m_0(S,τ_1))(1-π/π f_1(q_1(τ_1)) - π/(1-π)f_0(q_0(τ_1))) + q(τ_1)(f_1(q(τ_1)|S)/f_1(q_1(τ_1)) -f_0(q(τ_1)|S)/f_0(q_0(τ_1)))]
       ×[(m_1(S,τ_2) - m_0(S,τ_2))(1-π/π f_1(q_1(τ_2)) - π/(1-π)f_0(q_0(τ_2))) + q(τ_2)(f_1(q(τ_2)|S)/f_1(q_2(τ_2)) -f_0(q(τ_2)|S)/f_0(q_0(τ_2)))],

	and 
	
    Σ_3(τ_1,τ_2) = 𝔼[m_1(S,τ_1)/f_1(q_1(τ_1)) - m_0(S,τ_1)/f_0(q_0(τ_1))] [m_1(S,τ_2)/f_1(q_1(τ_2)) - m_0(S,τ_2)/f_0(q_0(τ_2))],

	respectively. 



	Let 𝒜_n = {(A_i^*,S_i^*, A_i, S_i): i =1,⋯,n}. Following the definition of M_ni and arguments in the proof of Lemma <ref>, we have 
	
    {W_sfe,n,1^*(τ)-𝒲_n,1(τ)|𝒜_n}
    d=   {∑_s ∈𝒮1/√(n)[∑_i=N(s) + 1^N(s)+n_1(s)(M_ni-1)(η̃_i,1(s,τ)/π f_1(q_1(τ)))  - ∑_i=N(s) +n_1(s)+ 1^N(s)+n(s)(M_ni-1)(η̃_i,0(s,τ)/(1-π) f_0(q_0(τ)))] |𝒜_n}
    
    	=    {∑_s ∈𝒮1/√(n)[∑_i=N(s) + 1^N(s)+n_1(s)(ξ_i^s-1)η̃_i,1(s,τ)/π f_1(q_1(τ)) - ∑_i=N(s) +n_1(s)+ 1^N(s)+n(s)(ξ_i^s-1)η̃_i,0(s,τ)/(1-π) f_0(q_0(τ))] + R_1(τ) |𝒜_n},

	where sup_τ∈Υ|R_1(τ)| = o_p(1) and {ξ_i^s}_i=1^n, s ∈𝒮 are sequences of i.i.d. Poisson(1) random variables that are independent of 𝒜_n and across s ∈𝒮. In addition, by the same argument in the proof of Lemma <ref>, we have 
	
    ∑_s ∈𝒮1/√(n)[∑_i=N(s) + 1^N(s)+n_1(s)(ξ_i^s-1)η̃_i,1(s,τ)/π f_1(q_1(τ)) - ∑_i=N(s) +n_1(s)+ 1^N(s)+n(s)(ξ_i^s-1)η̃_i,0(s,τ)/(1-π) f_0(q_0(τ))] 
    
    	=    ∑_s ∈𝒮1/√(n)[∑_i= ⌊ nF(s)⌋ + 1^⌊ n(F(s)+π p(s))⌋(ξ_i^s-1)η̃_i,1(s,τ)/π f_1(q_1(τ)) - ∑_i=⌊ n(F(s)+π p(s))⌋+1^⌊ n(F(s)+ p(s))⌋(ξ_i^s-1) η̃_i,0(s,τ)/(1-π) f_0(q_0(τ))] + R_2(τ) 
    ≡    W_1^*(τ) + R_2(τ),

	where  sup_τ∈Υ|R_2(τ)| = o_p(1). Because both W_sfe,n,2^*(τ) and W_sfe,n,3^*(τ)-𝒲_n,2(τ) are in the σ-field generated by 𝒜_n, we have 
	
    (W_sfe,n,1^*(τ)-𝒲_n,1(τ), W_sfe,n,2^*(τ), W_sfe,n,3^*(τ)-𝒲_n,2(τ)) 
    d=    (W_1^*(τ) + R_1(τ)+R_2(τ), W_sfe,n,2^*(τ), W_sfe,n,3^*(τ)-𝒲_n,2(τ)).

	In addition, note that {ξ_i^s}_i=1^n and {η̃_i,1(s,τ),η̃_i,1(s,τ)}_i=1^n are independent of 𝒜_n, therefore, W_1^*(τ)  (W_sfe,n,2^*(τ),W_sfe,n,3^*(τ)-𝒲_n,2(τ)). Applying <cit.> to each segment 
	
    ⌊ nF(s)⌋ + 1,⋯,⌊ n(F(s)+π p(s))⌋  or  ⌊ n(F(s)+π p(s))⌋+1,⋯,⌊ n(F(s)+ p(s))⌋
 for s ∈𝒮 and noticing that {η̃_i,1(s,τ)}_i=1^n and {η̃_i,0(s,τ)}_i=1^n are two i.i.d. sequences for each s ∈𝒮, independent of each other, and independent across s, we have, conditionally on {η̃_i,1(s,τ),η̃_i,0(s,τ)}_i=1^n, s ∈𝒮,   
	
    W_1^*(τ) ℬ_1(τ)
  
	with the covariance kernel Σ_1(τ_1,τ_2). 
	
	For W^*_sfe,n,2(τ), we note that it depends on data only through {S_i^*}_i=1^n. By Assumption <ref>, 
	
    W^*_sfe,n,2(τ)|{S_i^*}_i=1^n ℬ_2(τ)

	with the covariance kernel Σ_2(τ_1,τ_2). 
	
	Last, for W^*_sfe,n,3(τ) - 𝒲_n,2(τ), note that {S_i^*} is sampled by the standard bootstrap procedure. Therefore, directly applying <cit.>, we have 
	
    W^*_sfe,n,3(τ) - 𝒲_n,2(τ) = 1/√(n)∑_i =1^n (ξ'_i-1) [m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ))]+R_3(τ)

	where sup_τ∈Υ|R_3(τ)| = o_p(1), {ξ_i'}_i=1^n is a sequence of i.i.d. Poisson(1) random variables that is independent of data and {ξ^s_i}_i=1^n, s ∈𝒮. By <cit.>, conditionally on data {S_i}_i=1^n, 
	
    1/√(n)∑_i =1^n (ξ'_i-1) [m_1(S_i,τ)/f_1(q_1(τ)) - m_0(S_i,τ)/f_0(q_0(τ))] ℬ_3(τ),

	where ℬ_3(τ) has the covariance kernel Σ_3(τ_1,τ_2). Furthermore, ℬ_2(τ) and  ℬ_3(τ) are independent as Σ_2(τ_1,τ_2) is not a function of {S_i^*}_i=1^n. This concludes the proof. 
	




§ ADDITIONAL SIMULATION RESULTS




 §.§ DGPs

We consider the following four DGPs with parameters γ =4, σ =2, and μ which will be specified later. DGPs 1 and 3 correspond to DGPs 1 and 2 in Section <ref> in the main paper. 

	
  * Let Z be the standardized Beta(2,2) distributed, S_i = ∑_j=1^4{Z_i ≤ g_j}, and (g_1,⋯,g_4) = (-0.25√(20),0,0.25√(20),0.5√(20)). The outcome equation is 
	
    Y_i = A_i μ + γ Z_i + η_i,
 
	where η_i = σ A_i _i,1 + (1-A_i)_i,2 and (_i,1,_i,2) are jointly standard normal. 
	
  * Let S be the same as in DGP1. The outcome equation is 
	
    Y_i = A_i μ+γ Z_i A_i - γ(1-A_i)(log(Z_i+3)1{Z_i ≤ 0.5}) + η_i.

	where η_i = σ A_i _i,1 + (1-A_i)_i,2 and (_i,1,_i,2) are jointly standard normal.  
	
  * Let Z be uniformly distributed on [-2,2], S_i = ∑_j=1^4{Z_i ≤ g_j}, and (g_1,⋯,g_4) = (-1,0,1,2). The outcome equation is 
	
    Y_i = A_i μ+A_im_i,1   + (1-A_i)m_i,0 + η_i,

	where m_i,0 = γ Z_i^2 1{|Z_i|≥ 1} + γ/4(2 - Z_i^2)1{|Z_i|<1}, η_i = σ(1+Z_i^2)A_i_i,1 + (1+Z_i^2)(1-A_i)_i,2, and (_i,1,_i,2) are mutually independent T(3)/3 distributed. 
	
  * Let Z_i be normally distributed with mean 0 and variance 4, S_i = ∑_j=1^4{Z_i ≤ g_j}, (g_1,⋯,g_4) = (2Φ^-1(0.25),2Φ^-1(0.5),2Φ^-1(0.75),∞), and Φ(·) is the standard normal CDF. The outcome equation is 
	
    Y_i = A_i μ+A_i m_i,1 + (1-A_i)m_i,0 + η_i,

	where m_i,0 = -γ Z_i^2/4, m_i,1 = γ Z_i^2/4, 
	
    η_i = σ(1 + 0.5 exp(-Z_i^2/2))A_i_i,1 + (1+0.5exp(-Z_i^2/2))(1-A_i)_i,2,

	and (_i,1,_i,2) are jointly standard normal.


When π = 1/2, for each DGP, we consider four randomization schemes:

	
  * SRS: Treatment assignment is generated as in Example <ref>.
	
  * WEI: Treatment assignment is generated as in Example <ref> with ϕ(x) = (1-x)/2.
	
  * BCD: Treatment assignment is generated as in Example <ref> with λ = 0.75. 
	
  * SBR: Treatment assignment is generated as in Example <ref>. 

When π≠ 0.5, we focus on SRS and SBR. We conduct the simulations with sample sizes n=200 and 400. The numbers of simulation replications and bootstrap samples are 1000. Under the null, μ=0 and the true parameters of interest are computed by simulations with 10^6 sample size and 10^4 replications. Under the alternative, we perturb the true values by μ = 1 and μ = 0.75 for n = 200 and 400, respectively. We consider the following eight t-statistics. 


	
  * “s/naive": the point estimator is computed by the simple QR and its standard error σ_naive is computed as 
	
    σ^2_naive =    τ(1-τ) - 1/n∑_i =1^nm̂^2_1(S_i,τ) /πf̂_1^2(q̂_1(τ))+τ(1-τ) - 1/n∑_i =1^nm̂^2_0(S_i,τ) /(1-π)f̂_0^2(q̂_0(τ))
        + 1/n∑_i =1^n π(1-π)(m̂_1(S_i,τ)/πf̂_1(q̂_1(τ)) + m̂_0(S_i,τ)/(1-π) f̂_0(q̂_0(τ)))^2 
        + 1/n∑_i =1^n (m̂_1(S_i,τ)/f̂_1(q̂_1(τ)) - m̂_0(S_i,τ)/f̂_0(q̂_0(τ)))^2,

	where q̂_j(τ) is the τ-the empirical quantile of Y_i|A_i = j, 
    m̂_i,1(s,τ) = ∑_i =1^nA_i1{S_i = s}(τ - 1{Y_i ≤q̂_1(τ)})/n_1(s),
 
	
    m̂_i,0(s,τ) = ∑_i =1^n(1-A_i)1{S_i = s}(τ - 1{Y_i ≤q̂_0(τ)})/n(s)-n_1(s),
 and for j=0,1, f̂_j(·) is computed by the kernel density estimation using the observations Y_i provided that A_i = j, bandwidth h_j = 1.06σ̂_jn_j^-1/5, and the Gaussian kernel function, where σ̂_j is the standard deviation of the observations Y_i provided that A_i = j, and n_j = ∑_i =1^n {A_i = j}, j=0,1. 
	
  * “s/adj": exactly the same as the “s/naive" method with one difference: replacing π(1-π) in σ^2_naive by γ(S_i).
	
  * “s/W": the point estimator is computed by the simple QR and its standard error σ_B is computed by the weighted bootstrap procedure. The bootstrap weights {ξ_i}_i=1^n are generated from the standard exponential distribution. Denote {β̂^w_1,b}_b=1^B as the collection of B estimates obtained by the simple QR applied to the samples generated by the weighted bootstrap procedure. Then, 
	
    σ_B = Q̂(0.9) - Q̂(0.1)/Φ^-1(0.9) - Φ^-1(0.1),

	where Φ(·) is the standard normal CDF and Q̂(τ) is the τ-th empirical quantile of {β̂^w_1,b}_b=1^B. 
	
  * “sfe/W": the same as above with one difference: the estimation method for both the original and bootstrap samples is the QR with strata fixed effects.
	
  * “ipw/W": the same as above with one difference: the estimation method for both the original and bootstrap samples is the inverse propensity score weighted QR.  
	
  * “s/CA": the point estimator is computed by the simple QR and its standard error σ_CA is computed by the covariate-adaptive bootstrap procedure. Denote {β̂^*_1,b}_b=1^B as the collection of B estimates obtained by the simple QR applied to the samples generated by the covariate-adaptive bootstrap procedure. Then, 
	
    σ_CA = Q̂(0.9) - Q̂(0.1)/Φ^-1(0.9) - Φ^-1(0.1),

	where Q̂(τ) is the τ-th empirical quantile of {β̂^*_1,b}_b=1^B. 
	
  * “sfe/CA": the same as above with one difference:  the estimation method for both the original and bootstrap samples is the QR with strata fixed effects.
	
  * “ipw/CA": the same as above with one difference:  the estimation method for both the original and bootstrap samples is the inverse propensity score weighted QR.  




 §.§ QTE, H_0, π = 0.5



































 §.§ QTE, H_1, π = 0.5





















 §.§ QTE, H_0, π = 0.7























 §.§ QTE, H_1, π = 0.7




















 §.§ ATE, π = 0.5




























 §.§ ATE, π = 0.7











	
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