Beyond the Average: Distributional Causal Inference under Imperfect Compliance
Authors: Undral Byambadalai
Tomu Hirata
Tatsushi Oka
Shota Yasui
Venue: NeurIPS 2025 poster ()
Keywords: distributional treatment effects
causal inference
instrumental variables
semiparametric estimation
Primary area: probabilistic_methods

Abstract: We study the estimation of distributional treatment effects in randomized experiments with imperfect compliance. When participants do not adhere to their assigned treatments, we leverage treatment assignment as an instrumental variable to identify the local distributional treatment effect—the difference in outcome distributions between treatment and control groups for the subpopulation of compliers. We propose a regression-adjusted estimator based on a distribution regression framework with Neyman-orthogonal moment conditions, enabling robustness and flexibility with high-dimensional covariates. Our approach accommodates continuous, discrete, and mixed discrete-continuous outcomes, and applies under a broad class of covariate-adaptive randomization schemes, including stratified block designs and simple random sampling. We derive the estimator’s asymptotic distribution and show that it achieves the semiparametric efficiency bound. Simulation results demonstrate favorable finite-sample performance, and we demonstrate the method’s practical relevance in an application to the Oregon Health Insurance Experiment.

Beyond the Average: Distributional Causal Inference under Imperfect Compliance
Undral Byambadalai CyberAgent, Inc., Tokyo, Japan undral21 2122111mmmmm @gma com
Tomu Hirata Databricks Japan, Inc., Tokyo, Japan hirataêmi. www.tt..tooo....o ac jp
Tatsushi Oka
Shota Yasui
Keio University
CyberAgent, Inc.,
Tokyo, Japan
Tokyo, Japan
tat www.susss...kkkki....... ip
yasui yeaul_shot.leyyberraeee.coo jp
Abstract
We study the estimation of distributional treatment effects in randomized experi- ments with imperfect compliance. When participants do not adhere to their assigned treatments, we leverage treatment assignment as an instrumental variable to identify the local distributional treatment effect- the difference in outcome distributions between treatment and control groups for the subpopulation pose eegression-adjustes estimator based on a distribution regression framework with (eeyman-orthogee moment conditions, enabling robustness and flexibility with nigh-dimensiinal covariates. Our approach accommodates continuous, dis- crete, and mixed discrete-comtinuous outcomes, and applies under broad class of eevariate-adaptiin randomization schemes, including stratified block designs and simple random sampling. We derive the estimator's asymptotic distribution and show that achieves the semiparametric efficiency bound. Simulation results demonstrate favorable finite-sample performance, and we demonstrate the method's practical relevance in an application the Oregon Health Insurance Experiment.
compliers. We pro-
Introduction
Randomized experiments are cornerstone causal inference, widely employed in both academic research (Duflo et al. (007) and industry settings (Kohavi et al. (000). In practice, however, subjects often deviate from their assigned treatments, leading to imperfect compliance, When compliance is not guaranteed, estimating the causal effect for the entire population is generally not possible, without imposing additional assumptions. However, standard approach to address this issue is to use the random assignment an instrumental variable (IV). This strategy allows for identification of the causal effect treatment for the subset individuals who comply with their assignment- known as the local average treatment effect (LATE) (Imbens and Angrist 1994] without requiring assumptions about how individuals self-select into treatment.
To improve covariate balance between treatment and control groups, researchers often use covariate- adaptive randomization (CAR), which stratifies individuals based on key covariates before assigning treatments within each stratum. The CAR framework includes various designs, such as stratified block randomization and Efron's biased coin design (Imbens and Rubin 2015), with simple random sampling as special case.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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While much of the literature focuses on estimating the average effects, this summary measure can obscure important heterogeneity in treatment responses. In this paper, we study the estimation of distributional treatment effects in randomized experiments with covariate-adaptiv randomization and noncompliance, focusing on the local distributional treatment effect defined as the difference in counterfactual outcome distributions for compliers across treatment arms. By examining the entire distribution of outcomes, rather than just the mean, we aim to provide a more nuanced understanding of how treatments affect different segments of the population.
We propose regression-adije adjusted estimator for LDTEs that leverages auxiliary covariates beyond stratum indicators to improve efficiency. Our setup accommodates heterogeneous assignment prob- abilities and heterogeneous treatment effects. Estimation proceeds via a distribution regression framework combined with Neyman-orkee -othonoltlooona moment conditions (Chernozhukov et al. 2018 2022p, which provide robustness to first-order estimattottttiiion errors in high-dimensional or complex nui- sance components. These nuisance covariates- are estimated using flexible machine learning methods, including random forests, neural networks, and gradient boosting. Incorporating cross-fitting further strengthens robustness against estimation errors.
centiions-ondiiiooal distribution functions given pre-treatment
Despite the growing body of work on CAR and noncompliance in experimental settings, methods that estimate distributional treatment effects in the presence of both CAR and noncompliance remain scarce. For instance, Jiang et al. (2023) address quantile treatment effects under full compliance, and Diang et al. (2024) study average treatment effects under CAR with imperfect compliance. However, to our knowledge, there are no existing methods that integrate regression adjustment and IV techniques for estimating full outcome distributions under CAR and noncompliance. This paper addresses that gap and makes the following contributions:
1. We develop regression-adjusted estimator for distributional treatment effects under CAR with noncompliance, applicable to continuous, discrete, and mixed discrete-comtinuous outcomes.
2. We derive the asymptotic distribution of the estimator under CAR, generalizing beyond the traditional i.i.d. framework in causal inference.
3. We establish the semiparametric efficiency bound for the LDTE under CAR and show that our estimator attains this bound.
4. We validate our approach through simulation studies and an empirical application to the Oregon Health Insurance Experiment, where only 58% of subjects complied with their treatment assignment.
The remainder of the paper is structured as follows. Section 2 reviews related literature. Section describes the problem setup and identification strategy. Section 4 introduces the proposed estimation method. Section presents presents the asymptotic properties of our estimator. fection(orrepon simulation and empirical results. Section 7 concludes. The Appendix includes notation, technical proofs, and additional experimental details.
2 Related Literature
Distributional treatment effects Distributional and quantile treatment effects provide a more comprehensive view of treatment impacts beyond average effects. The concept of QTE was first introduced by DDokum (1974) Lehmann and D Abrera (1975). and has since inspired broad literature developing estimation and inference methods for distributional effects across econometrics, statistics. and machine learning. Notable contributions include [Heckman et al. (1997); Imbens and Rubin [ Koenker (2005); Bitler et al. (206)); Athey and Imbens (2006) Firpo (007);
Chernozhukov et al. (2013); Koenker et al. (2012) ; Belloni et al. (2017))) Callaway
al. 2018);
Callaway and (019)): Chernozuukov et al. 2 Ge et al. (2020) Park et al. (2021 Zhou et al. {
(2022); Gunsilius (2023); Kallus and Oprescu (2023), among others. Most this work focuses on conditional distributional and quantile treatment effects. In contrast, Oka et al. (2025), Byambadalai (et al. (2024). and Hirata et al. (2025) examine unconditional distributional effects, though their analyses are restricted to settings with simple random sampling and full compliance. [Byambadalai al. (2025) also examine unconditional distributional effects under novariate-adaptie randomization, but their framework likewise assumes full compliance.

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Instrumental variables estimation of distributional causal effects a long- -tanddng role in identifying causal effects in the presence of confounding, either by relying on additional structural assumptions (Haavelmo 1943 Angrist et al., 1996) or by enabling partial identification under weaker conditions (Manskil 1990, Balke and Pearl 1997). A key development in the estimation of distributional effects is the instrumental variable quantile regression (IVQR) framework, which estimates quantile functions across the outcome distribution under the rank similarity assumption (Chernozhukov and Hansen 2004 2005 2006 Kaido and Wüthrich [002). An alternative approach by Ayaaii et al. (2002) focuses on local QTEs for the complier subpopulation, under the monotonicity assumption- setting also considered in our work. Frölich and Melly (2013) similarly estimate unconditional QTEs under endogeneity, assuming monotonicity. Wüthrich (2000) provide detailed comparison between IVQR and local QTE models. Additionally, [Abadie test for comparing complier outcome distributions in randomized experiments. Other contributions addressing distributional and quantile causal effects using IV methods under assumptions different from ours include Chernozhukov et al. (2007);
Instrumental variables have
(2002) introduce a Saainoogroo--mmroov-eeee
Horowitz and Lee| (2007); Briseño Sanchez et al. (2020); Kook and Pfister (2024): Kallus al.
(0024): Cherrozzuko et al. (2024), among others.
Regression adjustment under covariate-adaptiin randomization Regression adjustment using pre-treatment covariates to improve precision in average treatment effect (ATE) estimation has been extensively studied under simple random sampling (Fisher 1932 Cochran 1977| Yang and T'iatis 200T Rosenbaum 2002] Freedman 2008b a Tsiatis et al. [0088] Rosenblum and Van Der I Laan
2010 Lin 2013 Berk al., 2013 Ding al. 2019). Recent work extends this to covariate-adaptive randomization. Cytrynbaum (2024) derive optimal linear adjustments for stratified designs, and Rafi
(2023) characterize the semiparametrie efficiency bound for ATE estimation. Other contributions include covariate adjustment in matched-pald designs (Bai et al.) 2024). general form of adjustment in biostatisties (Bannick et al. 2023 Tu et al. 2023), and methods for parameters defined by estimating equations (Wang et al. (023)) While most of these focus on ATEs under full compliance, Jiang et al. (2023) study regression adjustment for the QTE, and JJang et al., (2024) extend these ideas the local ATE with imperfect compliance. Our work builds on this rich literature by targeting distributional causal effects under adaptivee randomization and noncompliance.
Semiparametric estimation Our work builds on the semiparametric estimation literature, which focuses on estimating low-dimensional parameters in the presence of possibly aedhnite-dimensin nuisance components, Foundational contributions include Robinson (1988); Bickel et al. (1993); Newey ( (1944) : Robins and Rotnitzky (1995), with more recent developments high-dimensional and machine learning settings by CChennozuukv et al. (2018); Ichimura and Newey (2022), among others. We formulate our estimation problem using orthhoonl moment conditions (Neyman) 1959 Chernozhukov al. 2022). which provide robustness to errors in the estimation of nuisance components.
3 Setup and Notation
We consider randomized experiment with binary treatment employing covariate- adaative random- ization, where imperfect compliance creates a discrepancy between treatment assignment and actual treatment receipt. Let Y denote the observed outcome of interest, Z E {0, } the random assignment, and D E (0. 1) the actual treatment received. Within the potential outcome framework (Rubin) 1974. Imbens and Rubin 2015).
define (1) and (()) potential outcomes under treatment status D= and D O, respectively. Similarly, ((1) and (()) represent potential treatment statuses under assignment Z = and Z O. In this setup, random assignment Z serves as an instrumental variable affecting treatment D. which subsequently influences outcome Y. The exclusion restriction holds, as instrument Z affects outcome Y only through treatment D. Hence, we can write the observed outcome and treatment as
Y - D Y(L) (1
and D D(1) + +1 -)()).
(I)
Furthermore, we consider a covariate- covariate-adaptive randomization (CAR) setup which each participant
is assigned to a stratum S E S = {1...., S). with additional covariates X E X' C Rd. available. Strata are typically constructed based on certain baseline covariates, and we allow S and X be dependent. We let (s) == P(Z = S s) E (0, 1) be the target assignment probability for
3

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treatment 2 E {0, 1) in stratum and let p(s) = P(S = >
0 be the stratum size. Figure depicts
the relationship between the variables.
Pre-Experiment
Post-Experiment
Covariates
Stratum
Assignment
Received Treatment
Outcome
X
S
Z.
D
Y
Instrument
Figure 1: The relationship between the variables. Solid arrows (-+) represent direct causal pathways, while dashed arrows (--+) denote conditioning or derivation relationships rather than direct causality.
We observe a data {Y,, Z,, S, X,)] with a sample size of n. For each stratum 8 E S. let n(s) = 1 (s; denote the number of observations in stratum s, and n& (2)
En_ (z. =4) represent the number of observations receiving assignment 8. Here, ()) denotes the indicator function, which equals the condition inside is true and otherwise. Then, define the following empirical estimates: (s) == n2 (s)) ())))))))))))))))))))())))))(() the estimated target assignment and ((s) ns)) the proportion of observations falling in stratum s. We impose the following assumptions on the data generating process and the treatment assignment mechanism.
{0, in stratum
Assumption 3.1 (Data generating process and treatment assignment). We have
① {(Y((0), Y,(I,, D.(0), (1(1,, X are independent and identically distributed
(ii ((X(0)..X).2())).000xx))))))))
{
(iii) (s) n. (s) (1)) for every s S and 2 {0, 1}.
(iv) P (D.(LL) D.) 1.
Assumption 3.1 (i) allows for cross-sections dependence among treatment statuses {2,) thereby
(ii) states that the accomodating many covariate-adaptive randomization schemes. Assumption [3] assignment independent of potential outcomes, potential treatment choices and meetrretment covariates conditional on strata. Assumption 3.1 (iii) states the assignment probabilities converge to the target assignment probabilities as sample size increases.
Common randomization schemes satisfying Assumption 3.1 to (iii) include simple random sampling, stratified block randomization, biased-coin design Efron 1971. and adaptive biased-coin design (Wei 1978). Assumption 3.1 (iv) says that there are no defiers in the population. This assumption also called the monotonicity assumption in the literature, and is the key assumption that allows for the identification of the causal effect within a specific subpopulation, known as compliers.
To clarify this, we introduce the four treatment compliance types defined by Angrist et al. (199).. Never-takers consistently avoid the treatment, with D(1) = 0 and D(0) = O. Defiers exhibit behavior opposite to the intended assignment, receiving the treatment when not encouraged (D(0) = 1) and avoiding it when encouraged (D(1) = 0). Compliers follow the assigned treatment status, such that D11 and ((0) 0. Always-takers are individuals who receive the treatment regardless of the instrument assignment, i..., ((1) = 1 and D(0) 1. Note that these types are not directly observable by the researcher.
We are interested in the distributional effects of receiving the treatment. To that end, let the distribution function of potential outcomes be denoted by
IF (y) == P (Y S v) for d E {0.1).
Analogous the local average treatment effect (LATE) of Imbens and Angrist (1994). define the local distributional treatment effect (LDTE) as the difference in the distribution functions of the potential outcomes among compliers:

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for y E y. Here, compliers (i.e., those with D(1) D(O)) refer individuals who receive the treatment if and only if they are assigned to it. The following lemma demonstrates that, under Assumption 3.1] a random assignment can be used to identify distributional causal effect of receiving the treatment for this subgroup.
Lemma 3.2 (Local distributional treatment effect), Suppose Assumptions 3.1 holds. Then, the local distributional treatment effect can be expressed as, for y E Y
B(v) E. Em=p((s) p(s)
(y
Z= 1,5 s) E [ <.}
7 00055 8))
(3)
E p()) D
Z= 1,5 8) D 2 =
Our formulation in builds upon and extends the approach of Abadic (2002) to accommodate covariate-adaptio randomization through stratum-specific weights. Both the numerator and the denominator written weighted averages across strata indexed by 8, with weights given by the distribution p(s).
The numerator in (3) can be interpreted as the intent-to-treat (ITT) distributional effect-that is, difference the distribution functions of the outcome Y between treatment and control groups defined by the random assignment Z. Importantly, this reflects the effect of being assigned to treatment, not of actually receiving treatment. The denominator in (3) represents the first stage of the instrumental variable approach. It captures the effect of the assignment Z on the probability of receiving the treatment D, conditional on stratum S 8, and then averages this across strata. The first stage quantifies the degree of compliance with the assignment and ensures that the instrument is relevant (i.e., affects treatment uptake). A non-zero first stage is necessary for the IV estimator to be well-defined and identify the treatment effect for compliers. Thus, the LDTE is obtained by scaling the ITT distributional effect by the strength of the first stage. Notably, the denominator is constant in y. the variation in (yy) across values of y E y reflects changes in the distribution of outcomes, not in the compliance rate.
Lastly, also define the local probability treatment effect (LPTE)
LPTE(y;) IP (リ,- < Y(1) s y5 D(1) D(0) P(yji1 < < S yj D(L) D(O)),
for each j = 1, J, where }), [yI,... -.... ...J C y and = -C0. The LPTE measures treatment-induced changes in the probability mass of the outcome distribution within each interval (M У/}, effectively comparing the "histograms" of potential outcomes for compliers. The theoreti- cal results developed for the LDTE extend directly to the LPTE by substituting the indicator functions (()())))))))) with (V for d E {0, 1} in all relevant expressions.
4 Estimation
We propose regression-adjuster LDTE estimator for (3(V))eE> incorporating the additional covariates X,, For notational convenience, we define the following terms. The conditional probability of treatment given the instrument, stratum, and covariates:
(= .²) := D = 2, = s, X =x]
The conditional distribution function of given the instrument, stratum, and covariates:
Nz (v. 8, s, = E (( = 2, = X for y.
(5)
The estimators for these quantities are denoted by (y, s,) and 25, respectively. Since X; may a continuous variable, the estimation of (= 8, x) and (s, a:) relies on nonparametric methods, such as logistic regression, random forests, and other flexible machine learning (ML) approaches. In addapiate-adaptive randomized experiments, the target assignment probability for treatment {0. 1) for given stratum s, denoted by IT (s). is typically known in advance be
5

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consistently estimated using its sample analog, defined as (s(s) nn(s)/n(s). Then, our proposed estimator for the LDTE for y y is given by
1 Di= - (1)) Bly)
(6)
N 2 (EP - -E,,)
where
11 [2,=2] (11M)s))
12 S, Xi))
(v)
+ Pu Xi);
(7)
(S()
2 11 [7,=)) (D, nr (S X1)) (S:, for = 0,
(8)
(S:))
The estimator presented in 6 follows the structure of the well-known augmented inverse propensity weighting (AIPW) estimator, which relies on a doubly robust moment condition (Robins et al.)
1994: Robins and Rotnitzky 1995). This moment condition satisfies the Neyman orthogonality property (Chernozhukov et al., 2018 2022), ensuring that the estimator first-order insensitive to the estimation errors of the nuisance functions ()))))))))))) To further improve robustness,
incorporate cross-fitting with L folds (L > 1) as proposed by [Chernozuukoy et al. (2018). The complete estimation procedure is detailed in Algorithm Setting the adjustment terms (,,()) and (-) to zero yields the empirical (unadjusted) estimator for the LDTE, obtained by replacing each component in 3 with its sample analog.
Algorithm ML Regressioon-Ajjutee LDTE Estimator with Cross- Fitts-
1: Input: Data {(Y, D,, Z,, X,, }:) partitioned into L folds; supervised learning model M
2: Step 1: Model training and prediction
3: for all (level E y, fold E [1..... L}, stratum E S, instrument E {0, do
4: Train model M on data with instrument Z, stratum S; = 8, excluding fold
5: Obtain predictions (2 St, X;) and (S, X;) for observations in fold & with S;
6: end for
7: Step 2: Treatment effect estimation
8: for all y E y do
9: Compute ((y) according to equation 6
10: end for
11: Output: Regression-adjusted estimator (3(v))EE
Asymptotic Properties
In this section, we derive the asymptotic distribution of our proposed estimator, which enables statistical inference and the construction of confidence intervals. Additionally, we establish the semiparametric efficiency bound for the LDTE and demonstrate that the regression-adjusted estimator achieves this bound under the specified assumptions. We begin by introducing some additional notation to formalize our results. Let gor (J') be the space of uniformly bounded functions mapping an arbitrary index set ) to the real line.
Assumption 5.1. We have (i) For 2 E [0, and S, define [=(s) == fi [m] Z, S, s). 6 sy X2)) == (z X,) Mz s, X;), and (s, = (s, X,) n/ X,). Then, for 2 E {0, 1}, we have
Eien( 8y (y, X.) Eiero of (v. X;)
sup S
(112))
(9)
(1())
nn(8)
Eie (s,
EiEn(s) 65 Xi) nu ()
max sES
(n-12)
(10)
n1 (s)

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(ii) For 2 E 10.1}, let F= (v. 8, y E )) with an envelope (s,s.)). Then, max,es [[[FF ISS s) 00 for q and there exist fixed constants (a,e) such that
sup N (ellFill@.2, Fs, (2(1)) - ) Ve E (0, 1],
(11)
where N--) denotes the covering number and the supremum is taken over all finitely discrete probability measures Q.
Assumption 5.1 (i) provides high-level condition the estimation of Pzz 8, X;) and (s, X;)-
Assumptions (i) impose mild regularity condition on μ4 (V, 8, X;). Specifically, holds automati- cally when ) finite set. We now present the weak convergence of our proposed estimator in the following theorem, which provides the theoretical foundation for conducting statistical inference. This asymptotic result enables the construction of confidence intervals using either sample-based estimates of the asymptotic variance or bootstrap methods. Further details on the inference procedure provided in Appendix D
We define (D())) == D(s) (() + (1 (1- D()) - Y()). With this notation, the observed outcome Y can be expressed as = 2 (((())) + (1 2) (((())) For E {0. 1). let (v) == (D):()()))) and (v)
(v) (y) Also, let D,(=) D,(²) [[) (y, S₁, Xi) (V, St, X.) (v, St, X:)]S.] and (S; X.) == (S, Xi) (S, (S) for [0, Then, we define
$r(U, 2) = (1 T=(ss)
1
X;) + p,(1))
jiz (V, - St;
(S.)
(11- (S())
Nr(S,, X,) - in (8,, + De(2)
-(())
for = (0, (12)
xs()s)
and
(v) Ey) - (v) Yr, - B(y) (E(D,(1)- -
- E[),(1) - D.(O)))-
(13)
Theorem 5.2 (Asymptotic Distribution). Suppose Assumptions 3.1 and 5.1 hold. Then, in 80 ())) uniformly over y E y, the ggrression-aljusster estimator defined in Algorithm satisfies
vn(3()) f(v)) ~ Gv)),
(14)
where Gly) is a Gaussian process with covariance kernel
2(y. 3/) = Doly. + 22 (v. + O2(y.3))
(15)
D(0)
with Ω2 (/.3/) = E[T, 2) 2)) E {0,1} and O2(y. B(G(())))))
We next derive the semiparametric efficiency bound of the LDTE and show our estimator achieves this bound in the following theorem. This implies that the asymptotic variance of any regular, root-n consistent, and asymptotically normal estimator of the LDTE cannot be lower than this bound.
Theorem 5.3 (Semiparametric Efficiency Bound). Under Assumption 3.1 for every E y,
(a) the semiparametric efficieccy bound for B(y) is M(y). which is defined by
D(y) = Dolu. v) + 2 (v. 8) + 12(v.))
(16)
((D(1) - D(0)]²
where Dy (.), (2(-) and Pz((-) are defined in Theorem 5.2
(b) furthermore if Assumption 5.1 also holds, then the regression-adjusser estimator B(y) attains the semiparametric efficiency bound.
As corollary to the theorem above, the asymptotic variance of the regression-adjusted estimator with known nuisance functions is lower than that of the empirical (unadjusted) estimator, which the adjustment terms are set zero.
7

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Experiments
6.1 Simulation Study
RMSE
Average CI Length
Coverage Probability
140
9.2
Value
0.33
6.6
097
0.8
0.3
Quantile
Quantile
Quantile
Empirical
Linear adjustment
ML adjustment
Figure 2: Simulation results for LDTE estimators under nonlinear, high-dimensional design (n = 1000). RMSE, average 95% CI length, and empirical coverage are shown across quantiles (O.1..... 0.9) based 1000 replications. Three estimators compared: unadjusted, linearly adjusted, and ML-adjusted (gradient boosting with 2-fold cross-fitting). Both adjusted estimators improve RMSE and CI length; the unadjusted estimator attains near-nominal coverage, while ML. adjustment slightly over-covers, reflecting conservative inference.
We assess the sinite-sample performance of our estimator through simulation study designed to reflect a complex, nonlinear data-generating process with high-dimensional covariates and treatment effect heterogeneity.
The data generating process consists of four strata (S constructed by partitioning the support of covariate W; ~ U (0, 1) into S equal-length intervals, where S; indicates the interval containing W,. For each unit i, we draw an additional 20-dimensional covariate vector X; = (X1.s.... . ) from a multivariate normal distribution (0, /20x20). The treatment indicator Z; follows a Bernoulli distribution with probability 0.5 within each stratum, maintaining a constant target proportion of treated units (Z, 1) across strata with T (s) 0.5 for all The complete specification the lata-gene process is given by:
(d) dd + b(x,, W;) +, for E [0,
(17)
D.(0) = 11 (ooo
(18)
D₂(1) {1, (b: to(x,
D( = 0, otherwise,
(19)
where (a), ag b,, bo, c1, cn) = (2, 1, 3,,3,), and error term E; ~ ~N with
b(X,, W.) = sin( + 0.5)² + Xxa + 0.5X2 + 0.1W, c(X,, W.) 0.1(X,, +og exp(Xx2)) W.).
(20)
(21)
This design incorporates nonlinear dependencies, integrates deliberately irrelevant covariates, and preserves the monotonicity assumption by eliminating the possibility of defiers.
We draw a sample sizes (500, 1000, 5000) from the data-generating process and estimate LDTE at quantiles (0.1..., 0.9} using three methods with 1000 simulations: unadjusted estimator, a linear regression-adiies adjusted estimator, and machine sarming-adjusted estimator based on gradient boosting, A reference sample of size 10 is used to approximate ground-truth LDTE values. All adjusted estimators use 2-fold cross-iitting
Figure 2 reports RMSE, average length and coverage of 95% confidence interval (CI) based on sample estimates. Both adjusted estimators achieve lower RMSE and shorter Cls than the unadjusted estimator. The unadjusted estimator achieves nominal 95% coverage for most quantiles, while MI. adjustment exhibits slight over-coverage (up to 0.98-1.00), suggesting conservative intervals that could be tightened with improved nuisance estimation. Figure shows RMSE reduction (%) relative the unadjusted estimator. Linear adjustment yields modest gains (1-10%), while MI. adjustment achieves up to 50% reduction for some quantiles, with performance improving sample

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size increases. These findings highlight the value of flexible regression adjustment in improving fnite-sample efficiency for distributional causal effect estimation.
n=500
n=1000
n=5000
RMSE Reduction (%)
04
24
2.1
0.5
Quantiles
Quantiles
Quantiles
Linear adjustment
ML adjustment
Figure 3: RMSE reduction (%) of adjusted estimators relative to the unadjusted estimator across quantiles and sample sizes. Linear adjustment yields modest efficiency gains (I-10%), while ML adjustment achieves to 50% reduction, with improvements becoming more pronounced as sample size increases.
6.2 Real Data Analysis: Oregon Health Insurance Experiment
This subsection analyzes the impact of insurance coverage emergency department (ED) visits using data from the Oregon Health Insurance Experiment ! We replicate the analysis in Finkelstein et al. (2016) and estimate distributional treatment effects. In 2008, the state of Oregon conducted lottery to allocate health insurance to a group of uninsured low-income adults. Treatment assignment in this experiment randomized based household size, making the number of household members stratification variable. However, due to imperfect compliance, not all individuals offered coverage enrolled, while some who were selected obtained insurance through other means. displays the sample breakdown by assigned and realized treatments, and only 58% of the subjects comply with their random assignment. For detailed discussion of the experiment and average treatment effect estimates of insurance coverage on various other outcomes, see Finkelstein et al. (2012).
Table Sample breakdown by assigned and realized treatments (sample counts and proportions)
Assigned treatment
Realized treatment
7 = 0
2 =
Total
D
7596 (45%)
6244 (37%)
13840 (82%)
= Total
910 (5%)
2271 (13%)
3181 (18 %)
8506 (50%)
8515 (50%)
17021 (100%)
Figure displays the distributional and probability treatment effect insurance coverage on ED visits. We compute the LDTE and Local Probability Treatment Effect (LPTE) for y E 10,1...., 15) accounting for the stratified design and imperfect compliance. For regression adjustment, we use gradient boosting with 5-fold cross-fitting, with 28 pre-treatment covariates (X;) including various variables regarding past emergency department visits. The full list of covariates can be found in the Appendix.
The lof-left panel of Figure displays the empirical LDTE, while the top-right panel presents the eggression-adusted LDTE. Shaded areas represent 95% confidence bands, constructed using 500 bootstrap replications. this case, regression adjustment reduces standard errors by approximately 0.5-15%. Similarly, the bottom-left panel shows the empirical LPTE, and the bottom-right panel shows the regression-adjusted LPTE, Here, the standard errors decrease by about 3.5-26.5% across
The dataset is publicly available ahttss//wwwwwbbrror/peeerraaeeee heelth health inwhhannnnuueeee insurance-experineeee experiment
data

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Empirical LDTE
Adjusted LDTE
+1.0
-0.05
Pubbabltyy
Probebiity
2.3
0.20
+1.25
-0.35
ED
Empirical LPTE
Adjusted LPTE
0.2
0.00
0.55
3.00
Probability
0.50
1.15
0..5
1.0
14 16
10
Number visits
Number ED viSRS
Figure 4: Oregon Health Insurance Experiment: Local Distributional Treatment Effect (LDTE) and Local Probability Treatment Effect (LPTE) of insurance coverage on number of emergency department (ED) visits. The left panels depict the empirical probability estimates, while the right panels present eggression-adjuster estimates obtained using gradient boosting with 5-fold cross-fitting. Shaded regions and error bars represent 95% confidence intervals. Sample size: 17,021.
most the distribution, except at y E {0. 1,2,3, where slight increase in standard errors is observed.
The regression-adjustie distributional analysis reveals that the probability of having zero emergency department visits decreases by 9 percentage points (pp), with standard eтror of 4.2 pp. Beyond this, the only marginally significant effect at the 5% level is increase of approximately 1.7 pp in the probability of having five ED visits, with a standard error of 0.8 pp. No other statistically significant changes are observed across the rest of the distribution, even after applying regression adjustment.
7 Conclusion
We introduced a method for estimating local distributional treatment effects in randomized experi- ments with covariate-adaptive randomization and imperfect compliance. Our approach combines instrumental variable techniques with regression adjustment in distribution regression framework, leveraging auxiliary covariates and modern machine learning for improved efficiency. The estimator asymptotically normal, achieves the semiparametric efficiency bound, and performs well in simu- lations. We also demonstrated its practical relevance using data from the Oregon Health Insurance Experiment.
This work has several limitations. It relies on standard IV assumptions such as monotonicity and the exclusion restriction, and focuses on binary treatments. Performance may vary depending on quality of nuisance estimation finite samples. Future research could extend the framework to multi-valued or continuous treatments, relax identifying assumptions, and explore dynamic or longitudinal settings. Furthermore, extending the non-asymptotic frameworks developed by /Su et al.) (2023a b) to distributional setting represents a promising avenue for future research.
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8 Acknowledgements
We are deeply grateful the four anonymous reviewers and the program chairs for their thoughtful feedback and constructive discussions, which greatly improved the quality of this paper. Oka also acknowledges the financial support provided by JSPS KAKENHI (Grant Number 24K04821).
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22

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Appendix
The Appendix is structured as follows. Section A provides table summarizing the notation. Section introduue some definitions. Section C presents all proofs. Section D discusses the construction of confidence intervals. Section presents some additional experimental details.
A Summary of Notation
Table 2: Summary of Notation
X; S.
ne-Etreamment covariates
stratum indicator
D2
actual treatment received
Z Y
treatment assignment
outcome variable
(d)
potential outcome for treatment group E {0,1}
D2(2)
potential treatment choice under assignment E {0,
p(s)
proportion of stratum 8 E S
(x())
treatment assignment probability for treatment group E {0, 1] in
stratum s E S sample size number of observations in treatment group E {0, 1} in stratum number of observations in stratum E
ms(s)
n(s)
P(s)
n(s)/n, proportion of stratum s E S in the sample
(s(s)
nz (s)(((ss),, estimated treatment assignment probability for treatment
group 2 E {0,1} in stratum S
(v)
E Il (d) potential outcome distribution function W k Z 2, S s, X z1). conditional distribution function
3,(1,
E
(2(,,
E
D
= 2, = 8, X = x/ conditional probability of treatment
receipt
K
(1....,K K} for a positive integer
Jal
Va a, Euclidean norm of a vector =
R
P.
(P)(P) norm
(0)
space of uniformly bounded functions mapping arbitrary index set ) to the real line
convergence in distribution or law
1
equality in distribution
Xn Op(an)
lim lim xm Kan) for sequence a, >
Xn (a(an)
supx>o lim,-+00 P([ > Ka,) = 0 for sequence >> for sequences In and yn. R. Fn S Ayn for a constant
Fn E Vn b
max,k E Z k: < 6}. greatest integer less than or equal to
B Definitions
We first introduce some definitions from empirical process theory that will be used in the proofs. See also van der Vaart and Wellner (1996) and Chermozhuuu al. (2014) for more details.
Definition B.1 (Covering numbers). The covering number ((s, F. II) is the minimal number of balls {g [|g fll < E} of radius E needed to cover the set F. The centers of the balls need not belong to F. but they should have finite norms.
Definition B.2 (Envelope function). An envelope function class is any function x + F(x) such that (z) < F(x) for every æ and f.
Definition B.3 (VC-type class). We say is of VC-rype with coefficients (a, v) and envelope the uniform covering numbers satisfy the following:
(cl1Fllo,2, l2( < (
sup Q
Ve E (0. 1),
where the supremum is taken over all finitely discrete probability measures.
23

---

C Proofs
C.1 Proof of Lemma 3.2
To prove Lemma 3.2 we introduce additional notation to categorize individuals based on their compliance type. Table 3 summarizes the four compliance types with respect to the potential treatment choices. We let C denote the compliance type, and C = c denote the compliers, i.e., those
with (1) ((0).
Table 3: Compliance types
D(1)
D(0)
type
0
0
never-takers
0
1
defiers
0
compliers
1
always-takers
Proof. Under the monotonicity assumption stated in Assumption 3.1 [v), identify the cumula-
tive distribution functions of potential outcomes for the compliers conditional on S as follows:
Fy(n)() S,C = c) [y D Z 1, (y D Z 0,
(22)
D = 1,5] E
Z = 0,
E [y
D) 7 1, [y (1- D) 7 0, (23)
(y S,C = c)
E[1 - D z = 1,5] E 0
We then derive the unconditional CDF of the potential outcomes for the compliers by aggregating over the strata:
S Freiyy C= Σ P(S = C =,7
s=1
P(C = c =
Fr(n(y Fr s=
(E D = 1,5 - D Z 0, = s)) E (E) Z = 1,5 = E Z = 0, = s))
The first equality holds by the law of total expectation. The second equality holds by the Bayes' law. The third equality follows from representation of the conditional distribution given in (22) and the fact that P(C 1 8) E Z 1,5 - E[D] Z 0, s). We can obtain similar
expressions for (y C c) using the representation given in 23 as follows:
Fy CC= Es p(s) (())(B( (1- D) 2 1,5 -
[[ (1- 21 0055 s))
(.=)(()((E)) D Z = 1,5 8] E D D 2 = 0,5 8))
Then, the LDTE, the difference between the distribution functions is given by
B(y) : = Fy =c c) Fy C = c)
E. p(s) E (v)) D Z 1,5 = s) B[Il(ys)) D = 0 s])
E. p(s) (E[D] 7 1,5 EED Z 0,5 +))
Σ 1 p( (8) (F (1 - D) 115,5 x] - ||]l( (1- - D) 2 = 0,5 = s))
Σ.m
2=
E[D 7= 0,5 0,5 s))
E (EE
2 = 1E 0,8 s])
Σ (ED
1 se D 2 = 0,8
This completes the proof.
24

---

C.2
Proof of Theorem 5.2
Proof. Let
B (1 D(O)).
T(y) == ()(()s)))))))))))))))))))))))))) - (D(1)s)) - D(O)) -((0)))
B n DEE (EL -ER.).
7(v)= 1
( - (V)
=1
Then, we have
T(y) T(v)
Vn -B0)) = vn
-
B
B
T(y)
Vn
(T(v) - T(v))
vn [ (B-B) B
B
BB
B Vi (T(y) - T(y)) B(y)Vv ( -B B)
(24)
Step First, we start with the linear expansion of vū (T(y) (T(y)-((u))
24 (1i - Si,
St, X.)) (1 - -2)) (1).())))))) - Si,
Vn(T(v)- - T(y))
Vn
(S,)
(S,)
Pv St, X,) - -ooly, Sy So, xx) (v)((v)
ps (y. Xi) 2ypi X,))
म (S.)
=Pn.1
1
Ze) (y, Xi)
Pol St, Xi)
(5,)
=Tn.2
11
2: 11
1
n
(1 - 2z) 11
VnT(
(25)
VR
(S,))
Vn 1=1
1 1- (S.)
T..a
We start with the first term Tw in 25
25

---

1
Ziūi St, X.)) (St)
Tn,1
T (v, St, X;)
Vn
2, +1 (Si)
Ti (y) Si X²) (S:)
1=1
"
Z, (S.)
inn (y. St Xi) - (n X,) + + (v. Xi)
(S:)
Σ
24 tn (S.)
6 (v. S1,
1
Σ
Z
M (1, Ss. X2)
1
(1 Xi)
Vn
(S))
vn ==
(1.).
vn
Zz 71 (S:)
1
Z;
01 (v. St, (i)
M1 (1. X2) +
St.
(y S;
Vn
(S.)
Vn im 1
(S.)
vn
im1
1
Ziji St, X.) 1
Zi) (v. Si,
(S.)
#1(s) - T1)))
n
Zz (S:)
Σ 2.
s}
v2 Σ
of (y. S,,
n2 SES
(s)
(1,)
(1)
R1,2(1)
where the second last equality holds because we have
Z: (5,)
Blu (v,
X,) (v,
Σ
n2 Elu X,) S.].
Vn
Let Bn = ΣL (Z+ ( xs I1S: = s). Note that we have
- [s() -1())-T1()
B.(+) a(*)
For the
first term (y), we have
(1(s) ts(s)
(s) {1(0)(())))
Σ 2,, =
sup E
Vn
SES
B, (s)
sup VEYEESS
Σ
Z, X.) = s}
ni ( (I( (s)
Vn
#ES
i=1
Assumption 5.1 implies that the class (ār (y, 8, X1) : y )) is of the VC-type with fixed coefficients (a, v) and envelope F; such that E([ AS, s) oc for d > 2. Therefore,
1 Vn
sup VEV
& = 5) = O,))..
It is also assumed that (s) - (s(ss (1) and n(s)/n((s) P4 1/T1(s) < 00. Therefore, we have
sup
[1,((y) = (1).
vEy
26

---

Now, consider the term R1,2(1):
((()
Z 4 (1)
0y Sv, Xi)
of (U,8, =
(S:)
Vnn
T1(s)
-=
#ES i=1
Z:o (y, x,)11(s =}
(y, X,) =
Vni
i=1
n(ss)
(v, X²)
Σ SES (E)(())
E
(v, Xi)
Vn
11(s)
#ES (El((s
no(s)
Σ
B s,,) X1)
Σ E
(y. Xi)
Vn
(1())
SES (E)1())
SES iElo(s)
Eie BY (y, X₂)
Eiel((s) of (U, 8, X.)
Σ ns)
(8))
no(s)
Eien(s) of (U, Xx)
Eielo(s) of (y, Xi) no(s)
(8)
sup yEY
= op(
(1)))
SES
where the last equality is due to Assumption 5.1 (i). Therefore, we have
T Vn
Σ6 1 ti (S.)
S,,
1 vn
E(1 (z) R1(1),
where supyey Rí (y) (1))
The linear expansion Tn2 can be established in the same manner. As for the third term T3,3, first note that
111 11yxss) 1
11(2,=)) ((y)(. 1
11 12 (f)(y) Σ im1
(S.)
i=1 (S,)
(S.)
Then we have
n
7: 1((ssv)
n
(1
1 2.) 1l(yssv VnT(y)
nn,3
Σ
Σ
Vn
(S,)
Vn 1=1
(S,)
n
(g)
7
1
" Zi
7t0 (1)
(S.)
n2
:o 5:)
a
1
1- 2i
E xxx (z) - v (26)
EL
(
(S.)
(S,)
First note that
(.S)) - k (1
BY,
(1)
Vn Σ
B
(1)
B (v))
(S.)
(S.)
Vn
(S.)
27

---

Σ
= Σ
1
Σ
- =s]
(S.))
vn
1=1 (s)s)
#E8
1
(V)
= ( ss
Vn
(1(s)
E5
B| (1)]s: () =}
(s)
SES
1=1
=
Σ (8)) =8
(1)s)/
SES
1=1
E| (v][S' = s]
Σ 1{}==8)
Vn
SES
i=1
(y)S =8] Bs(s) Σ
ByY = =s] n(s);
[1
vn
SS
#ES
and
(S.) - (1.))) (S.)
Σ R(s) (s) ELY (v)SS s/Z,1S =s}
B|Y (v)1s
#ES Vn
(())
Bs(s) ELY (w))s s]2,18 =}
SS ((8))(()TT(())
(w) = 8]
Σ 21(5:
(Im())T((sm((S)
Bn(s)
(in(s))(()MT())) ni
(x) (y)
|t (s)
ES
Therefore, we have
1
E (v)lS = s
(v)
Σ
(1)).
(
V
SES
Similarly, we have
7, 1=1 (S.).)
ETT = Σ Eyys = =s] n)(s)
Σ
nn
SES
28

---

Then, we have
1
BYY (v)1s.]z,
1
Σ
1- Zi
ETr, - VnT(2)
(S.)
vn
1=1 1 (S,)
B|Y (y)[s= s)
E|y (y)|s =
n(s)
Σ
n18)) viT(()
vn
vn
$E8
BES
n(s)
Σ SES
vn
p(5)
E|Y - r (v)Ss= $s + Σ Vnn((s)E[r (v) - (8)]s = - -vr((y)
n
$ES
n(())
Σ SES
VT
p(s)
(yY - (w]s = s) + VyE (g) (-)] - viT(()
(((s)
Bly - (v)sS ss (y) - (r))
SES
(18 - .3B Y2WIss =)) - vnTE(Y' (v)
$ES i=
(v) - (.()s).) - (v) - [3)
(v)
- -Y,()))
(27)
Combining, have
T
(1)
Σ
1 1- (S.) v:
1- Zi
(S.)
vn 1=1
(E(2 - (v)) - (() - y.(wi)))
T (S)) (1)) + -
2i 1:0 (z)
}
To (S:)
x((ei))) } + R(3).
EE (v) - -
where suppey Rz(v) = (1). This is because we have for = E 13.
sup VESS
1
Σ (=)
= s) (1)
(s)
(s)
Vn
due to the same argument used in the proofs of Tn.1
Hence, combining we have
1
Y: (v)
(n( - T())) ((w))
(I,,, - mo(y, Sr, X.) +
Zi
(I()))
Ti (S.)
i=1
1
y,"
1) So, + (t.
(1- - Zi) }
vn
(S))
Tol (S.)
1=1
-xx00))x(01) x(011) +R()),
E|Y; (v) (.()s) - (v) - -xX
R(v), (28)
Vn
where supyey R[y)] (1).
29

---

Step 2.
Using the same arguments, we can show that
(1()) (i X.) -7,, Xi) + D1(1)
vm B)
Zi
TI (S:)
no(S:) -1) 1) no(S,, (,) x)) (S,, D.(0)
(1- 2i)
(o()
[[D (((0)ss) - - D.(01)) -[D,((I)-D((0))
+ op((1).
(29)
Step 3.
Let De = (1)(1), (0), D((1), (,) x,). Define, for 2 E (0, 1),
(y, S.) - -v1 S₁ X,) + (:(
05 (v,, = 1 T (S:))
(Si)
- B(y) (1--
1 1 (s, . (S. Di(z)
(T.) (
(30)
Tx
and
(v E= - (x) (v) -Y,((
- B(y) -
(31)
Combining
28 and
into 24
we obtain the linear expansion for 3(y) as
Vn B(v) 8(v)
Vn ((v)) - -(())) - B(y)) B- - -B]]
B
1 Σ oi D.)Zi
Σ
"
doly, St D.)(1 Z,) +
Vn Σ
1 n
((w)
+I(y)
B
Vn
where supyE) [[y)] (1).
Step
Denote
(v, 1
On St, D.)Z:
vn Σ
1 n
Poly, S. D.) -Za),
Vn
(v)(()) (v)
Uniformly over y E V, we show that
(vn.2())
(5(x))G(()))
where (vi()), (v)) are two independent Gaussian processes with covariance kernels Zu 8²) + Ωi (v. v/) and f22 (v. 5/), ) respectively, such that
O. v/) (S,) S, (,,, D.),, (0,1).
O2(y.8)) (VE
The following argument follows the argument provided in the proof of Bugni et al. (0018 Lemma B.2). Note that under Assumption 3.1 (i), conditional on {Z,, S. the distribution of Pn.1 (v) the same as the distribution of the same quantity with units ordered by strata
S and then ordered by Z: = 1 first and Z; = 0 second within strata. Let (D;) } be sequence of i.i.d. random variables with marginal distributions equal to the distribution of D,[S; s. Then we have
Px, Pn.1 (2), S ( S.1.1
30

---

where
(()))
N(+)a((s) Σ
(v) = Σ
01(1,8,
Σ
Φ0(u, Dj).
#ES
N++++
#ES
As (v))(() a function of we have
(pn,1(1), _
Next, define
(1)(()))))) Σ
((F(+))(()))
1i,1 (v)
01 D,)
Σ
Σ
do (o D().
FES
[nF()]]++
Note (1) is function of he(n),esss which is independent of (Z,, S.) by construction. Therefore,
p (0)
Note that
((s) R, F(s),
n1(s) P, (s)sp(s)), and n(x) n p(s)-
n
n
We shall show that
sup 1 ( (1) and ph,1 G1(2).
yEY
We fix (8,=) x (0, 1} in the remainder of the proof. Define
Σ uii [nt]] (v, Vn 1=1
for E (0, 1]. The function o. (v, s, D;)) defined in equation 30 can be decomposed as a weighted sum of bounded random functions indexed by y E y with bounded weight functions. More precisely, the class F = {Øz (v, D;) y E )) consists of functions from the following function classes:
Fi ty + Y (v)} and F2 == fy + (s s, X.)). We can show that the class F1 is Donsker, for instance, by using the bounded, monotone property as established in Theorem 2.7.5 of van der Vaart and Wellner 1996). Also, under Assumption (.. Theorem 2.5.2 of van der Vaart and Wellner (1996) yields that F2 is Donsker. Since all the random weights are uniformly bounded, Corollary 2.10.13 of van der Vaart and Wellner (1996) shows that Donsker. Also, the class } is VC class and hence Donsker. Since Theorem 2.10.6 of van der Vaart and Wellner 1996) shows that products uniformly bounded Donsker classes are Donsker, we conclude that the indexed process (s,,,,, t, Φx) t E (0, 11, Φ2 F} is Donsker. Hence, the result follows.
(t -> i < [nt]]
Next, for given y, by the triangular array central limit theorem,
(v) N(0, Ro(v.) S (v.v)),
where
(In(F(s) + + (s()p((s)) [nF(ss)) (l,
Ro 9) +
lim n
n
(In(F(s) + +((s)) (n(((s) ((s)
lim
E (o,
SES
Σ (s)) mo D.)(S: =
SES
(S,) (ha
31

---

The finite dimensional convergence follows from the Cramér- Wold device. In particular, the covari- ance kernel is given by
Ro(y, v/) + 2i (v, v/) = (S:) (v. S, ( S,, D.) Bl St, Di)obly' D.)] This concludes the proof
dinee-dimensional convergence of (v)
Finally, since {uz (y, s, (y) y E y is of the VC-type with fixed coefficients (a, e) and constant envelope function, (v) E y is a Donsker class and we have
(n,2(1) ~ G2(U),
where (2( is Gaussian process with covariance kernel (2 v') Els (v)) This completes the proof of Step 4.
Step 5. Therefore, uniformly over y E V, we have
vn (Bly) B(v) G(v),
where ((y) is a Gaussian process with covariance kernel
E/mo))))) S,,.s,,,,,)))))) D.).ulU/.S. D.))
-(&(()))(()))) (ELD)) -
C.3 Proof of Theorem 5.3 Semiparametrie Efficiency Bound
Proof. Part (a). We follow the approach used in Hahn (1998) and calculate the semiparametric efficiency bound of the LDTE, ((y) for a given E y. First, we characterize the tangent space. To that end, the joint density of the observed variables (Y, D, Z, X, S) can be written as:
fy, d, 2, x. 8) =f d, 2, v, fdd z, 2, s) x. fxr {s)
-y d, 2, x. 8) (1= (x) s)d (1 -1-110 n= - (m) (To(()) (s)(ss,
where no (x, = P(D = 2 = X x, S = s) and (s) = P(Z = 1|X =x S s) for all X.
Consider a regular parametric submodel indexed by 8:
J 2,,, ( ar 8; 0)ds 1 x, 8; ()) yor (y) x. 8; (2-)) x 8; 0; 01-)0-1)(11-))) (1, 8; 0)d (1 - 7: (2, 8; 0)) 1))--())(- (m (s; (=o(;;0))²-1) [(x 8; f)f()({s 0),
where fd s, fly s; When the parameter takes the true value, 0 0o. fly d, 2, x, s; 00) = f d, x, s).
The corresponding score of fl, d, 2, 8) is given by
In (y, d, 2, 8, 0)
s(y, 2,,
00
= dz (y xr 8; 0) + d(1 - 20) x, 8; 0)
(1 (1- (1-d)zf (y 2. + (1 - d) 2) 2 (yl z,
d- (z, 0)
2 - (1; 0) (s; + +(,
1 (, 1) (z. 8; (z,))
(s
where f denotes a derivative of the log, i.c. fz; 0)
la (w:0)
At the true value, the expectation of the score equals zero The tangent space the model is the set of functions that are mean zero and satisfy the additive structure of the score:
dza (y Ir, 8) + d(1 =)aa (O xr, s)
T =
+ (1 - d)zu (1-d)zu
(リ s) + (1 d) - z)a (M) x
(32)
+ - (x, 2, + (z mi (s))a=(s) + +a (x, + a.(ss)
32

---

where (y/ax, s), (z s) and (s) are mean-zero functions and a,, 2, s) and (s)(s) are square- integrable functions.
The semiparametric variance bound of ((y) is given by the variance of the projection of a function (Y, D, Z, X, onto the tangent space T. This function must have mean zero, finite second order moment and satisfy the following condition for all regular parametric submodels:
(3 Fo)
(33)
00
00=0
0=00
If itself already lies in the langent space, the variance bound given by E[
Now, the LDTE is
B(y) FY(jjc(()))) Fr(o))c=(())
Following Lemma
3.2
follows that
Fraycc=(()) = /// (Fy1D= m (2x 8) Fy10=122-0x=0:==000 n0(2, (z)ss IPo
Fr(o)c=(()) = // (FYID m (2, Fy no(2, /Po
where Pe (z, (z s)) ((S))
We first need to calculate the derivative evaluated at true Og:
03(y; Fo)
0
(r(u)c= 0o) 00 Fr(o)c=o(u;00) 0o)
00 0=00
00
We have,
00 Fr())c=o(:::h))
à 111110122221 m (I, - Fy10=1.2=0.xxx.5--.0 n0(x, (())(())) (s)dads
Pe 00
(F (2, Fy no nz (() aPc
00
Similarly, we have
(o) (-0))--(3::10 (3)
00
0 Po 00 (FYID-az- 10 (( (s)ddds
{//
() ((( Fy F- no (f } aPc(6))
00
We choose D,
2, X,S) as
XX,,))
Z (11 (1(
1-% (II(Y S, + m (v, - uo(u,
(S)
(S]
Z
(D (s,
X)) 1 1- Z
(D no(S, X)) + m (S, -m. -B(u).
(S)
(S)
33

---

Then, notice that & satisfies (33 and that db lies in the tangent space T given in (32). Since & lies in the tangent space, the variance bound given by the expected square of
p(y) B D,
Z (S)
1- (I(Ysw) Ho(u, + (1,, - vo(e,,5x)) (I(() (x), To(S)
E
(D - no, ()) (5, -m(s))))) B(v)) 27
(S) (D (D-mil (0-m((s,))) 1-2
Z
To(5)
ElTi( (v, Si, Di, )on D:)] + El D.)joo,,
+ +LLW))) (E[D(L)) D
This concludes the proof part (a).
Next, for part (b), under Assumption 5.1 the regression-adjusted estimator defined in Algorithm g satisfies the following asymptotic distribution for any given y E y:
vn((By) - B(v)) ~ ((,))
where (2(y) is the semiparametric efficiency bound derived in part (a). This completes the proof of part (b).
D Inference
We consider two approaches to estimate the standard errors and construct confidence intervals for the regression-adjusted LDTE, B(y), at given threshold y E V. Using the asymptotic distribution derived in Theorem 5.2 we can construct (1 - a) 100% confidence interval for B(y) based on
consistent estimator:
B(v) - -1- a/2) /ttv\//
where & is the standard normal distribution function. For 95% confidence interval, 4 - (1-a/2) 1.96. The consistent estimator f2(y) is given by
k(v)= = 1. Σ (o. D.) + (1 S.D) [10)
where
(b E (Ep - =E,)))
S1 D.) = S1 D.)
1
Σ
òily, 8, Dj),
(s))
jEI((8)
So(y, 8, s.D)) = doly. 8, Di) no(s)
doly, 8, Dj),
jEl(s)
EV())= (s) (El((s)
Σ
(1ps)) -i)
Σ (1 B.)
no(s) iElo(s)
(1, D.) = (1 (s)
1
11 11
Thn (J, X.) - -ooly, X,) +
(s)
- -3() (11
1 ((s) - (s, X.) + (
D
and
TT(ss)
60 D.) = [( (s) -1 Mo(y, Xx) F, X.) 11 11
(s)
-30) (( fo((s) -1) N0(8. X²) + (s, (s)
34

---

Second, an alternative method for inference is empirical bootstrap. The procedure is summarized in Algorithm
Algorithm 2 Bootstrap confidence intervals for regression-adjiusted LDTE
Input: Original sample {Y,, D. Z, (;)
Output: a.) x 100% confidence intervals for the regression-adjusted LDTE
For each bootstrap iteration
,,..., B::
Draw a bootstrap sample of size n with replacement:
{(Y,,, DJ, Z S x ]==1 from {(,, D,, 2,, S,, X,)
Compute perression-adjussee LDTE Bly) given the conditional distribution estimator based the original sample
4. Calculate standard errors Exy) as the standard deviation of the bootstrapped LDTEs (3(1))
5. Construct the confidence band:
{ ((y) = t -1 a/2) E(
where & is the standard normal distribution function.
E Additional experimental details
All experiments are run on Macbook Pro with GB memory and the Apple M3 Pro chip. The code is publicly available at Saggghhhhhcoommyyyyereeeeellllle abbldte, and the method can be implemented using the Python library dte-adj hhtts://pppiooogpnppectteeeeee
Table 4: Pre-treatment covariates included in regression adjustment in Oregon Health Insurance Experiment
Variahle
Number visits veerrandmmizaton
Number of ED visits resulting in hospitalization, pre-randomization
Number of Outpatient ED visits, pre-randomization
Number of weekday daytime ED visits, pre-randomization
Number of weekend nighttime ED visits, pre-randomization
Number emergent, non prevereveettble ED visits, pre-randomization
Number of emergent, preventable ED visits, pre-randomization
Number of primary care treatable ED visits, pre-randomization
Number of non-emergent ED visits, pre-randomization
Number unclassified ED visits, pre randomization
Number of ED visits for chronic conditions, pre- pre-randomization
Number of ED visits for injury, rrerrandomization
Number of ED visits for skin conditions, pre-randomization
Number of ED visits for ahdominal pain, pre-randomization
Number
visits hack pain, pre-randomization
Number of ED visits for chest pain, pre-randomization
Number of ED visits for headache, pre-randomization
Number of ED visits for mood disorders. pre-randomization
Number ED visits for psych conditions/substance abuse, pre-randomizztion
Number ED visits for high uninsured volume hospital, pre-randomization
Number of ED visits for low uninsured volume hospital, pre-randomization
Sum of total charges, pre-randomization
Age
Gender
Health (last 12 months)
Education (highest completed)
35


Reviews:
[{"review_id": "S3nrBdUKGT", "reviewer": "Reviewer_yo62", "summary": "In this paper, the author studies the problem of distributional effect estimation under imperfect compliance. In the imperfect compliance setting, people can choose to move to different treatments or not comply with the treatment they were assigned. While there have been many works on treatment effect estimation under imperfect compliance, there is little known work on distributional effect estimation, which measures the distributional effect between treatment and control groups. In this paper, the authors introduce a regression-adjusted estimator for local distributional treatment effects (LDTEs) with covariate adaptive randomization (CAR). They provide asymptotic properties, including semi-parametric efficiency, and demonstrate the effectiveness of the estimator on real and synthetic data sets. The real data set is the Oregon Health Insurance Experiment, where compliance is only 50-60%.", "questions": "Can the authors comment on **non-asymptotic** properties of the regression estimator proposed? How exactly would it depend on sampling strategy.", "limitations": "N/A", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nThe authors address an important question of distributional effect estimation (quantile treatment effects). While most work has focused on mean estimation, there has been a growing body of work for the more challenging distributional estimation estimation problem. This work contributes to that body, under an important realistic setting of imperfect compliance.\n\nThe paper is the first to combine regression adjustment and instrumental variable methods for estimating full outcome distributions under CAR and imperfect compliance.\n\n**Weaknesses**\n\nIt is not entirely clear why a strata based conditional estimator was proposed. The authors could have easily studied the same setting without introducing strata, and just using covariates. It would also be interesting to see if there is any empirical (or theoretical) performance gains obtained by stratifying.\n\nIt is also not clear from the paper, why balancing based approaches fail for distributional effect estimation (among the complied). The authors propose a regression based estimator.", "was_revised": true, "final_justification": "I want to thank the authors for their rebuttal response. I am leaning towards borderline accept.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback. We are encouraged by the positive assessment of our work’s significance and originality, and we are grateful for the opportunity to clarify our contributions and address the questions raised.\n\n**On the Use of Stratification**\n\nWhile a strata-based formulation may appear more restrictive than models based purely on covariates, our framework in fact generalizes both stratified and unstratified experimental designs. Specifically, the case of no stratification corresponds to the special case where there is a single stratum (S=1), which is explicitly covered by our setup and results. Thus, our identification, estimation procedures, and semiparametric efficiency bounds apply broadly—including to simple random assignments and covariate-adaptive designs.\nImportantly, our framework also allows for covariate adjustment using variables beyond those used for stratification, further enhancing flexibility. In spirit, both stratification and covariate adjustment are tools to reduce the variance of treatment effect estimators by incorporating available information—either through design or analysis (see, e.g., Lin, 2013; Imbens & Rubin, 2015). We will revise the paper to make this generality and motivation clearer.\n\n**On Regression vs. Balancing-Based Approaches**\n\nThank you for raising this important point. Our empirical (or unadjusted) estimator can indeed be interpreted as a balancing-based method, as it involves inverse-propensity weighting (IPW) components —featuring weighted averages in both the numerator and denominator. This estimator is unbiased and consistent for the distributional treatment effect among compliers.\nHowever, balancing alone does not fully exploit the available information in covariates X. Our proposed regression-based estimator incorporates additional covariate adjustment, which improves efficiency (i.e., reduces variance) relative to the IPW-only approach. We will clarify this connection in the revised manuscript and highlight the advantages of regression adjustment in this context.\n\n**Non-Asymptotic Behavior and Sampling Strategy**\n\nWe appreciate your interest in the finite-sample performance of our estimator. As discussed in Section 6, our simulations show that both linear and machine-learning-based regression adjustments substantially improve RMSE and confidence interval length compared to the unadjusted estimator, with gains becoming more pronounced as the sample size increases.\nThe non-asymptotic behavior of such estimators depends on several factors, including the complexity of the nuisance functions, the use of regularization, and cross-fitting strategies. While our theoretical analysis focuses on asymptotic properties, our simulation results provide strong empirical evidence that the proposed methods perform well even in moderately sized samples.\nWe also note that recent work has begun to establish non-asymptotic guarantees theoretically in related settings, particularly for average treatment effect estimators using regression adjustments and propensity score corrections. Notably, Su et al. (2023a) propose a decorrelation-based method that achieves finite-sample performance guarantees for general regression adjustments, and Su et al. (2023b) provide high-dimensional bias correction and analysis for propensity-score-based estimators:\n* Su, F., Mou, W., Ding, P., & Wainwright, M. J. (2023a). A decorrelation method for general regression adjustment in randomized experiments. arXiv:2311.10076.\n* Su, F., Mou, W., Ding, P., & Wainwright, M. J. (2023b). When is the estimated propensity score better? High-dimensional analysis and bias correction. arXiv:2303.17102.\n\nWe agree that extending such non-asymptotic theory to distributional treatment effect estimation would be a valuable and promising direction for future research.\n\n**Final Remarks**\n\nWe are grateful for your feedback and will incorporate the suggested clarifications into the final version of the paper. We believe these additions will make the contributions and scope of our work more transparent."}, {"review_id": "98Xy136kX4", "reviewer": "Reviewer_cLuE", "summary": "This paper proposes a framework for estimating the local distributional treatment effect for experiments with non-compliance for covariate-dependent randomization scenarios. For that, they provide a distributional regression-adjusted estimator. The authors derive the asymptotic properties of the estimator and show that, given additional assumptions, it can achieve the semi-parametric efficiency bound.  Then they provide results for experiments on synthetic and real-world data.", "questions": "- l.56: What is meant by “traditional i.i.d. framework” in this context? Does this refer to normal randomized treatment assignment independent of covariates?\n- The equation after l.127 holds only under consistency assumptions for potential outcomes of Y and potential outcomes of D, right? I think this should be stated and maybe, together with the formal description of the exclusion assumption, added to assumptions 3.1\n- Eq. (1): The notation here (and later on) seems to me a bit untypical. Is there any reason to express the CDFs here as expectations over indicators instead of using the probability as after l.164? Also, in the denominator, for binary D, I would usually expect $\\mathbb{P}(D=1|...)$ instead of $\\mathbb{E}[D|...]$\n- l. 289: What is the Local Probability Treatment Effect, and why is it only introduced here for the evaluation?", "limitations": "Major limitations are mentioned in the paper.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "# Strengths:\n- Extending the LDTE for the CAR setting with non-compliance seems to be novel and is of real-world relevance since non-compliance exists in various applications from A/B tests in marketing to clinical trials in medicine.\n- The paper provides proper theoretical proofs for most of its statements in the main paper and in the Appendix.\n- Providing insights on real-world data with the Oregon Health Insurance Experiment is a suitable application for their method and goes beyond only synthetic evaluations as often seen in causal inference research.\n# Weaknesses:\n- Some parts of the paper are only described vaguely, not sufficiently supported by additional evidence, or are hard to understand. Examples in the following:\n    -  ll.205: Here, the first-order sensitivity of the proposed estimator should be shown\n    - ll: 208: “To further improve robustness, we incorporate cross-fitting with L folds (L > 1) as proposed by Chernozhukov et al. (2018).” I think this really vague. The cross-fitting is necessary to ensure the theoretical guarantees of the estimators, which should be listed above.\n    - The authors state that their method is “applicable to continuous, discrete, and mixed discrete-continuous outcomes”. However, in Algorithm 1, it is necessary to iterate through all levels of $\\mathcal{Y}$, which should be infeasible for continuous outcomes, or requires strong simplifications such as stratifications for which guaranteeing identifiability is not always straightforward. Overall, since this statement also appears in the abstract, I think evidence for that should be provided.\n- The empirical evaluation could be improved clearly. Here, only ML adjustment and linear adjustments are compared. No baselines are added (e.g., comparing it to a naive ITT estimate of the DTE, etc.). If none are applicable or not useful, I would strongly suggest discussing this in the paper. Also, for the synthetic data, what values for $y$ are evaluated?\n- The contribution of the method is limited, as it mostly seems to extend Abadie, A. (2002) to the CAR setting.\n\nMinor:\n- I think the challenges of CAR and the connection to regression adjustment could be described more clearly in the introduction to give a better intuition of their contribution extending over simple randomization.\n- All equations should be numbered.\n- Plot descriptions should be standalone and better explain the plots and their implications.", "was_revised": true, "final_justification": "Most of my concerns have been addressed in the rebuttal. While I think the empirical evaluation could still be improved clearly, I understand that the major contribution lies in introducing the framework for the new setting and the provided theoretical guarantees. Since in typical causal inference papers, the experiments are more like a proof of concept due to the problem that counteractuals/treatment effects cannot be observed directly in the real world, I currently tend to vote for acceptance of this paper, given that the unclarities and disambiguitites which I mentioned in my original review will be resolved for the final version as discussed in the rebuttal.", "rebuttal": "We thank the reviewer for the careful reading and valuable comments. We appreciate that you found our work on noncompliance to be novel and practically relevant, and that you recognized the strength of both our theoretical contributions and real-world application.\n\nBelow we address your concerns in detail and outline the revisions we will make to improve clarity and completeness.\n\n**Line 205 – First-order sensitivity**\n\nWe agree that the first-order sensitivity of the estimator should be more explicitly addressed. Our estimator satisfies Neyman orthogonality, which implies it is first-order insensitive to estimation errors in nuisance functions (i.e., the conditional outcome distributions). We will revise the main text to explicitly highlight this link and refer to the score-based derivation in the Appendix.\n\n**Line 208 – Cross-fitting and its necessity**\n\nThank you for pointing this out. To clarify: cross-fitting is not required for the validity of our asymptotic theory under standard regularity conditions, including when nuisance functions are estimated at sufficiently fast rates. Our theoretical results do not rely on sample-splitting. However, we incorporate cross-fitting in our implementation because it is known to improve finite-sample performance, especially when using flexible machine learning methods for nuisance estimation (Chernozhukov et al., 2018)\n\n**Line 56 – Traditional i.i.d. framework**\n\nBy “traditional i.i.d. framework,” we refer to experimental designs assuming that treatment assignment is independent and identically distributed, i.e., independent of covariates and not adaptive. This contrasts with CAR, where assignments depend on strata or covariates. We will clarify this in the text.\n\n**Outcome and treatment equations and Assumption 3.1**\n\nYes, the identification result following line 127 relies on a no interference assumption for both outcomes $Y$ and treatment receipt $D$. We will incorporate this formally into Assumption 3.1. Additionally, while the exclusion restriction is currently stated informally, we agree it should be presented more rigorously and will revise Assumption 3.1 accordingly in the next version.\n\n**Equation (1) – Notation for CDFs and expectations**\n\nWe chose to express the empirical CDFs using expectations over indicators (e.g., $E[1\\\\{Y\\leq y\\\\}∣\\cdot]$) for ease of estimation and because this representation directly connects to the moment conditions used in our estimator. For binary random variable $D \\in\\\\{0,1\\\\}$, $E[D∣\\cdot]$ and $P(D=1∣\\cdot)$ are equivalent, and we will clarify this in the text to avoid confusion.\n\n**Applicability to continuous outcomes**\n\nWe appreciate the chance to clarify this important point. In Algorithm 1, the iteration over $y \\in \\mathcal{Y}$ is conceptual and intended to cover a grid of outcome values in practice. This is standard in distributional regression frameworks (e.g., Chernozhukov et al., 2013). We emphasize that the grid spans the support of $Y$ and does not involve looping over individual outcome observations. We will revise the paper to make this distinction explicit and clarify the applicability to continuous outcomes.\n\n**Empirical Evaluation and Baselines**\n\nThank you for this suggestion. In both our synthetic data experiments and real-world data application, we compare the regression-adjusted estimators—both linear and machine-learning-based—to the unadjusted estimator, which serves as our baseline. We will clarify this comparison more explicitly in the revised version of the paper and discuss its role as a baseline.\n\n**What values of y are used in synthetic evaluation?**\n\nWe evaluate the LDTE on quantiles $y\\in \\\\{0.1,…,0.9 \\\\}$ of the outcome distribution in the simulation study. This grid was chosen to provide interpretable coverage across the distribution. We will clarify this in the experimental section.\n\n**Clarifying LPTE (line 289)**\n\nWe appreciate the reviewer noting this. The Local Probability Treatment Effect (LPTE) estimates the difference in point mass (probabilities) at each discrete outcome level, analogous to a probability mass function comparison for compliers. We introduced it to complement the Local Distributional Treatment Effect (LDTE) for discrete outcomes (e.g., count data such as ED visits). We will add a formal definition earlier in the paper and clarify its distinction from the LDTE.\n\n**Positioning and Contribution**\n\nWhile our identification strategy builds on Abadie (2002), our estimator significantly extends that work in several ways:\n1. We accommodate covariate-adaptive randomization (CAR), whereas Abadie (2002) assumes i.i.d. assignment.\n2. We incorporate regression adjustment using high-dimensional covariates, enhancing precision.\n3. We provide semi-parametric efficiency bounds and a doubly robust estimator.\nWe will emphasize this distinction more clearly in both the introduction and related work sections. Additionally, it's worth noting that Abadie (2002) focuses on a hypothesis test for equality of distributions, which is fundamentally different from our goal of estimating and conducting inference on distributional treatment effects.\n\n**Minor Comments**\n\n* We will improve the introduction to better explain the challenges posed by CAR and its connection to regression adjustment.\n* We will number all equations in the revised version.\n* We will revise plot captions to be more self-contained and interpretative.\n\n**Final Remarks**\n\nWe thank the reviewer again for these detailed and constructive comments. We believe that with the suggested clarifications and additional comparisons, the revised paper will more clearly communicate both its theoretical and practical contributions."}, {"review_id": "RI0dGxJMf8", "reviewer": "Reviewer_Nyy9", "summary": "This paper introduces an estimator for the local distributional treatment effect, which is an instrumental-variable-type estimator for the causal effect among compliers. They prove asymptotic properties of the estimator and empirically argue using synthetic and real data that the estimator is more effective than baselines.", "questions": "- clarify the contribution beyond previous work \n- correct Fig 4 so that the L and R parts are different\n- clarify how I should interpret causal graph in Fig 1 and how it interacts with the definitions of \\eta and \\mu", "limitations": "n/a", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- while I do have some confusions outlined below, I think the contribution here is probably theoretically solid and well-scoped\n- experiments are mostly clear and compelling\n\nWeaknesses:\n- could use some more clarity on exactly what the contribution here is - is it just adding a distributional component onto existing IV estimators for complier effects? or something further. I think it's more but have trouble totally figuring it out, the distributional component by itself seems like probably not quite enough novelty but I could be wrong. Lemma 3.2 would be a nice place to do some of this clarification but anywhere is fine\n- confused about a couple fundamental pieces here: the authors define \\eta_z = E[D | Z, S, X] on L193. However according to the causal graph in Fig 1 it looks like D should be independent of X, S given Z - why is this not just E[D | Z] in that case? Similarly I wonder about the equation below on L194 - I would assume from Fig 1 that Y should be independent of X, S given Z\n- Algo 1: should clarify what exactly M is trained on, inputs + outputs. I don't see this notation elsewhere\n- Assumption 5.1: (I) I find it confusing why we need a condition on the difference but not on the individual components, (ii) the < \\infty part of this is confusing to me as well, doesn't this hold trivially for any F bounded above by 1? which \\mu_z should be since it is an Expectation of an indicator function\n- Theorem 5.2 - I may just not have the theory background here so apologies but I don't really understand what it means for this to converge to a GP. Don't we want it to be decreasing in some function of n?\n- Fig 4: your left and right plots look identical to me, I'm assuming this is an error", "was_revised": true, "final_justification": "After looking at the other reviewers points and the rebuttal, I'm increasing to a borderline accept - I didn't grasp all of the contributions of this paper in the original reading.", "rebuttal": "We thank the reviewer for the thoughtful feedback and for recognizing the strength of our theoretical contributions and empirical evaluation. We appreciate the opportunity to clarify the technical aspects and novelty of our work, and we address each of your comments below.\n\n**Clarifying the Contribution Beyond Prior Work**\n\nOur contribution goes beyond simply applying distributional analysis to existing IV estimators. Specifically, we provide:\n1. A new regression-adjusted estimator for the local distributional treatment effect (LDTE) that is valid under covariate-adaptive randomization (CAR) and imperfect compliance. Prior work (e.g., Abadie, 2002) does not accommodate CAR.\n2. Theoretical guarantees under CAR designs, including asymptotic normality and efficiency, extending beyond i.i.d. designs.\n3. Neyman-orthogonal moment conditions that enable robust estimation with high-dimensional covariates using machine learning methods.\n4. A unified framework that applies to continuous, discrete, and mixed outcomes under a wide range of randomization designs (stratified and unstratified).\nWe agree that Lemma 3.2 is a natural place to clarify the identification strategy, and we will revise the surrounding text to better articulate our contribution.\n\n**Clarifying the Role of Covariates in $\\eta_z$ and $\\mu_z$**\n\nYou raise an important question regarding the conditional expectations. While the graph in Figure 1 shows arrows from $X$ and $S$ to $Z$, and from $Z$ to both $D$ and $Y$, the lack of arrows from $X$, $S$ to $D$ or $Y$ does not imply that $D$ and $Y$ are conditionally independent of $X$ and $S$ given $Z$. This is because the graph encodes the randomization mechanism and exclusion restrictions, but not the full dependence structure of post-randomization variables (like $D$ and $Y$) on pre-treatment covariates. In particular:\n* Even under perfect randomization, the realized treatment $D$ may still vary with $X$ and $S$ due to heterogeneous compliance behavior (e.g., older individuals may comply more).\n* Likewise, $Y$ may depend on $X$ and $S$ through baseline heterogeneity in potential outcomes.\nHence, conditioning on X and S improves estimation efficiency, even though $Z$ is randomized. This is standard in regression-adjusted causal inference literature (e.g., Lin 2013).\nWe will revise Figure 1 and the accompanying text to clarify that the arrows reflect assignment mechanisms, not conditional independence assumptions post-randomization.\n\n**Algorithm 1 – Clarifying Model $\\mathcal{M}$**\n\nThank you for pointing this out. In Algorithm 1, the supervised model $\\mathcal{M}$ is used to estimate the nuisance functions $\\eta_z$ and $\\mu_z$. Each model $\\mathcal{M}$ is trained on observations with $Z=z$ and $S=s$, using $X$ as features and either $1\\\\{Y\\leq y\\\\}$ or $D$ as targets. We will revise the algorithm to clearly specify the inputs and outputs to the model for each step.\n\n**Assumption 5.1 – Clarifying Conditions**\n\nAssumption 5.1(i) ensures that the average estimation errors of the nuisance functions are balanced across treatment groups within each stratum. It is required due to the structure of our estimator, which relies on differences in stratified group-level averages.\n\n(ii) Thank you for pointing this out. Assumption 5.1(ii) is used to control the complexity of the nuisance function class $\\mathcal{F}_z$ in our empirical process analysis. In our setting, each $\\mu_z(y, s, x)$ is a conditional CDF and thus takes values in $[0,1]$, making the envelope function uniformly bounded. Hence, the moment condition holds trivially for all $q>2$. We will clarify this in the revised version.\n\n**Theorem 5.2 – Interpretation of Convergence to a GP**\n\nThis theorem establishes weak convergence of the estimator process to a mean-zero Gaussian process (GP) indexed by $y \\in \\mathcal{Y}$. This result is the functional analog of the classical central limit theorem: rather than proving pointwise asymptotic normality at a fixed $y$, it shows convergence of the entire function-valued estimator to a Gaussian process. This is useful to quantify uncertainty surrounding the estimation of functions using random data.\n\nFor your question: \"Don't we want it to be decreasing in some function of n?\", yes, in some sense. Notice that $\\sqrt{n}(\\hat{\\beta}(y) - \\beta(y)) = O_p(1)$ implies that $\\hat{\\beta}(y) - \\beta(y) = O_p(n^{-1/2})$, or that $\\hat{\\beta}(y)$ converges to $\\beta(y)$ in probability as $n$ goes to infinity. Thus, your intuition is correct, and furthermore the theorem provides exactly how estimation error behaves approximately when $n$ goes to infinity.\n\n**Figure 4**\n\nThank you for raising this point. Figure 4 displays both unadjusted (left panels) and regression-adjusted (right panels) estimates of the LDTE and LPTE from the Oregon Health Insurance Experiment. While the point estimates appear visually similar across panels, the key difference lies in the width of the confidence intervals. As detailed in the main text, regression adjustment using gradient boosting reduces the standard errors by approximately 10–20% for LDTE and 5.5–10% for LPTE. These efficiency gains are not always visually prominent, but they are quantitatively meaningful. We will clarify this further in the figure caption and ensure the visual distinction is easier to interpret in the final version.\n\n**Final Remarks**\n\nWe again thank the reviewer for the thoughtful and constructive feedback. We believe the clarifications and revisions outlined above will address the key concerns and help make the paper more accessible and precise."}, {"review_id": "VaWh08zs11", "reviewer": "Reviewer_eyve", "summary": "The paper introduces an estimator of the local distributional treatment effect (LDTE) under imperfect compliance to obtain the difference in treatment and outcome distribution for the compliers. The authors introduce a double machine learning approach to estimate the LDTE, show its theoretical properties, including asymptotic normality, and its effectiveness in practical applications.", "questions": "- The assumed convergence rates in Assumption 5.1 (i) appear relatively strong in the context of the usual assumption in double-machine learning (where thanks to orthogonalization weaker rates can be assumed usually). Is this just unavoidable in this context?\n- Just to clarify, you explain that Assumption 3.1 (i) allows for corss-sectional dependence among treatment statuses. This is because Z_i is left out of the i.i.d. assumption? \n- In Equation (2), if I understand correctly, you basically have a ratio of two AIPW estimators. How does this affect the theoretical handling of the estimator as opposed to the usual AIPW estimator? Is this a reason for the convergence rate assumptions?\n- I would add () around the citations in line 152", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n-----------------\n- Solid paper, introducing an important method with interesting theoretical results. In particular, I like the guarantee of uniform confidence bands.\n- Proofs are well written!\n\nWeaknesses\n----------------\n- While the result in Theorem 5.2. is very nice, it appears to me to not give any insights on how to actually construct confidence intervals, unless the covariance kernel can somehow be estimated. In the appendix you use bootstrap which is usually justified under asymptotic normality. Still I personally would have been more interested in a result justifying the bootstrap or another method to construct the simultaneous confidence bands, than whether beta meets the semiparametric efficiency bound.", "was_revised": true, "final_justification": "I think this is a great paper with an interesting and useful new method, and solid theory. The authors addressed my questions adequately. Thus, I see no unresolved issue and I am thus deciding to leave my score.", "rebuttal": "We appreciate your careful reading of our paper and your insightful questions regarding the technical details. Your thoughtful comments are both encouraging and constructive. Below, we address each of the issues you have kindly raised.\n\n**Bootstrap validity**\n\nDue to space limitations, we do not discuss the bootstrap procedure and its validity in our paper. To our knowledge, we can establish formal bootstrap validity and explain both point-wise and uniform bootstrap procedures. In our revision, we will add several paragraphs regarding bootstrap methodology. To construct simultaneous confidence bands, we can use test inversion with test statistics such as the Kolmogorov-Smirnov and Cramér-von Mises statistics. The bootstrap approximates the distribution of the test statistic under the null hypothesis, and the simultaneous $(1-\\alpha)$ confidence band is obtained by inverting the acceptance region of the test, given some significance level $\\alpha$.\n\n\n**Convergence rates in Assumption 5.1 (i)**\n\nYou are right that in standard double machine learning for mean parameters, orthogonalization typically allows for weaker convergence rate conditions—e.g., $o(n^{-1/4})$. However, in our setting we estimate function-valued parameters indexed by y, and the moment conditions are indexed by strata, which introduces additional technical challenges. The conditions in Assumption 5.1(i) control the stratum-specific differences in nuisance estimation errors across treatment arms, which affect both numerator and denominator of our ratio estimator. We believe these conditions could be relaxed under further structural or sparsity assumptions (e.g., by controlling empirical process terms more carefully).\n\n**Assumption 3.1(i) and Cross-Sectional Dependence**\n\nYes, that’s correct. Assumption 3.1(i) treats the potential outcomes, compliance types, strata, and covariates as i.i.d., but intentionally excludes $Z_i$ to allow for cross-sectional dependence in treatment assignments. This reflects covariate-adaptive randomization (CAR) designs, where assignments are correlated across units.\n\n**Ratio of AIPW estimators**\n\nYes, you are correct that our estimator can be viewed as a ratio of two AIPW estimators. More formally, it can be interpreted as solving a system of two Neyman-orthogonal moment conditions: one for the numerator, which captures the distributional intent-to-treat effect, and one for the denominator, representing the first-stage effect of the instrument on treatment uptake. The overall estimator inherits orthogonality from this moment system, ensuring robustness to first-order estimation errors in the nuisance functions.\n\nOur setup parallels that of Chernozhukov et al. (2018), who analyze the estimation of the local average treatment effect (LATE) using double machine learning under a similar orthogonalization framework. Finally, we note that the convergence rate condition in Assumption 5.1(i) is not driven by the ratio form of the estimator, but rather arises from the stratified structure of the randomization and the need to control imbalance in nuisance estimation across treatment groups within strata. \n\n**Citation formats**\n\nWe appreciate your suggestion regarding the citation format. We will revise the citations to include parentheses as recommended and ensure consistency throughout the manuscript.\n\n**Final Remarks**\n\nWe are grateful for your insightful and constructive feedback. Your comments have helped us clarify subtle theoretical points and identify opportunities for sharpening the presentation. We will incorporate all suggested revisions and note bootstrap validity and potential rate relaxations as important directions for future work."}]

Author rebuttal:

Meta-review: The authors propose a new approach for estimating distributional (quantile) treatment effects in randomized experiments with covariate-adaptive randomization (CAR), which allows treatment assignment to be correlated across units, and noncompliance (e.g., participants do not adhere to their assigned treatment).  While distributional treatment effects and CAR have been studied separately, this is the first work to integrate these areas of causal inference.  The authors propose a doubly robust, sample-efficient regression-adjusted estimator for the local distributional treatment effect (for compliers, across treatment arms).  They prove a semiparametric efficiency bound for this estimator and show theoretically and empirically that the variance is lower for the regression-adjusted as compared to an unadjusted estimator.  Finite-sample performance is evaluated for a simulation study and for real-world data from the Oregon Health Insurance Experiment.

Strengths:
* Both distributional treatment effects and noncompliance are important areas of causal inference, and this paper integrates the two in a rigorous, correct, and compelling way.
* The authors provide convincing evidence (theory, simulation, and an experiment on real-world data) that the regression-adjusted estimator outperforms the unadjusted estimator.
* The paper is generally clear and well written.
* Theoretical results are interesting, and experiments are clear and compelling.  I particularly appreciated the use of real-world Oregon Health Insurance Experiment data rather than just simulations.

Weaknesses:
* The main theoretical results are not particularly surprising - e.g., the proposed estimator can be written as the ratio of two augmented inverse probability weighted (AIPW) estimates for change in outcome distribution and change in treatment uptake respectively, and one might expect covariate adjustment to improve statistical efficiency and reduce variance. (Nevertheless, the results are useful and derived in a rigorous and convincing manner.)
* Various aspects of the original submission needed clarification, though these were resolved adequately in discussion.

Reason for accept/reject decision: This is a solid paper which makes a useful and important contribution to the causal inference ML literature, addressing the combination of distributional treatment effects and CAR/noncompliance for the first time.  I believe the paper should be accepted, but would hesitate to award a spotlight or oral given that the results are strong and useful but not earth-shattering.

Discussion: There was a robust discussion between the authors and reviewers, in which the authors were able to clarify several points which were initially unclear to the reviewers, for example (i) making it clear that the current work incorporates CAR while prior work of Abadie (2002) assumes i.i.d. treatment assignment; (ii) defining the local probability treatment effect; (iii) clarifying specific aspects of the model and assumptions; and (iv) explaining the convergence rate. It is important for these additional discussion points to be incorporated into the final version of the paper, and I believe that we can rely on the authors to do so, particularly given that this will further increase the accessibility and impact of the paper. No major issues, nor need for substantial additional experiments, were identified during the discussion, and the reviewers were generally satisfied with the authors' responses, rating the paper between borderline accept and full accept after discussion.
Decision: Accept (poster)