Distributionally Robust Treatment Effect

Ruonan Xu; Xiye Yang

arxiv: 2512.12781 · v2 · submitted 2025-12-14 · 💰 econ.EM

Distributionally Robust Treatment Effect

Ruonan Xu , Xiye Yang This is my paper

Pith reviewed 2026-05-16 22:26 UTC · model grok-4.3

classification 💰 econ.EM

keywords distributionally robust estimationtreatment effect predictionWasserstein distancepartial identificationFrechet boundscausal inferencepolicy evaluationheterogeneity

0 comments

The pith

A distributionally robust predictor for treatment effects in new settings preserves the sign of the source average effect but shrinks it toward zero according to the degree of heterogeneity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to predict how a treatment will work in a new location or time period when only data from an earlier setting is available. It minimizes the worst-case mean squared prediction error over all distributions that lie inside a Wasserstein ball centered on the observed source distribution. Because the joint distribution of potential outcomes is never observed, the authors replace the unidentified joint with its sharp Fréchet bounds and obtain an explicit closed-form predictor. The resulting rule keeps the same sign as the source average treatment effect yet pulls the magnitude toward zero; the amount of shrinkage rises with the amount of treatment-effect heterogeneity present in the source data. The method comes with consistency, asymptotic normality, and a two-step inference procedure for the estimated bounds.

Core claim

The minimax optimizer of the worst-case mean-squared-error prediction problem over a Wasserstein neighborhood around the source distribution is given by the source conditional treatment effect multiplied by a shrinkage factor that depends on the Wasserstein radius and on the Fréchet bounds of the unidentified joint distribution of potential outcomes; this predictor necessarily preserves the sign of the source average treatment effect while attenuating its magnitude in proportion to treatment-effect heterogeneity.

What carries the argument

Wasserstein-ball minimax optimization whose sharp value is obtained by replacing the unidentified joint distribution of potential outcomes with its Fréchet-Hoeffding bounds.

If this is right

The bound estimators are consistent and asymptotically normal, permitting standard inference.
A two-step procedure yields valid confidence intervals for the robust predictor.
The degree of shrinkage is explicitly tied to observable treatment-effect heterogeneity in the source sample.
The same construction applies to any policy whose target population differs from the source by a bounded Wasserstein distance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could be used to decide whether to roll out a program whose source estimate is positive but whose robustness-adjusted value crosses zero.
Choosing the radius parameter might be guided by observable differences between source and target covariates.
The method supplies a concrete alternative to assuming full transportability of conditional effects across sites.

Load-bearing premise

The Wasserstein ball around the source distribution correctly captures every relevant possible target distribution that could arise in a new location or time period.

What would settle it

Collect data from a new location whose distribution lies inside the calibrated Wasserstein ball yet produces an average treatment effect whose sign is opposite to the source estimate; the robust predictor should then fail to match that sign flip.

Figures

Figures reproduced from arXiv: 2512.12781 by Ruonan Xu, Xiye Yang.

**Figure 2.** Figure 2: Minimax Optimizer under Heterogeneous Treatment Effect [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: Robust Prediction for Different q might not be the most interesting case [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

**Figure 4.** Figure 4: Population Prediction τ DR , τp, and τo diction, respectively, using either Neyman variance bounds or sharp variance bounds. For the left panel of Figures 5 and 6, the sample size is 445. There, the sample predictions are close to the population prediction but with certain gaps, especially for the upper bound. Increasing the sample size by tenfold in the right panels leads to sample predictions aligning mu… view at source ↗

**Figure 5.** Figure 5: Lower Bound of the Best Prediction 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 200 400 600 800 1000 1200 1400 pop upper sharp upper Neyman upper (a) q = 2, n = 445 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 200 400 600 800 1000 1200 1400 pop upper sharp upper Neyman upper (b) q = 2, n = 4450 [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗

**Figure 6.** Figure 6: Upper Bound of the Best Prediction 34 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison of 95% Confidence Intervals For the sample size n = 4, 450, we also examine the IM confidence intervals based on sharp variance bounds and Neyman variance bounds, with and without the Bonferroni correction. The results are plotted in [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗

read the original abstract

Using only retrospective data, we study the problem of predicting treatment effects for the same treatment/policy implemented in a different location or time period. We propose a distributionally robust estimator that minimizes the worst-case mean squared error for the prediction of treatment effect over a class of distributions defined by a Wasserstein neighborhood around the source distribution. Because the joint distribution of potential outcomes is unidentified, the problem is inherently one of partial identification. We characterize the sharp upper and lower bounds of the minimax optimizer by exploiting the Fr\'echet class of distributions consistent with the marginal distributions of potential outcomes. The resulting predictor preserves the sign of the average treatment effect under the source distribution but is shrunk toward zero, with the degree of shrinkage depending on the extent of treatment effect heterogeneity. We establish consistency and asymptotic normality of the bound estimators, develop a two-step inference procedure, and discuss the choice of the robustness parameter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a Wasserstein DRO estimator for treatment-effect prediction under shift that shrinks the ATE toward zero while keeping its sign, using Fréchet bounds on the unidentified potential-outcome joint.

read the letter

The core contribution is a minimax predictor that trades off source-sample fit against worst-case mean squared error over a Wasserstein ball around the observed distribution. Because the joint of potential outcomes is unidentified, they replace the inner problem with sharp Fréchet bounds on the marginals and show that the resulting estimator is a shrunk version of the usual predictor, with the shrinkage factor rising in treatment-effect heterogeneity. They prove consistency and asymptotic normality of the bound estimators and sketch a two-step inference procedure. That combination is not a routine extension of existing DRO or partial-ID work, and the sign-preservation property is a clean, usable feature for policy applications where one wants to avoid sign flips under modest shifts. The abstract is clear on the setup and the main result, and the free parameter (the Wasserstein radius) is at least discussed explicitly. The main soft spot is whether the Fréchet extremal joint is guaranteed to lie inside the Wasserstein ball defined on the observed (X, Y) law. The abstract asserts that the bounds are sharp for the constrained problem, but without the derivation it is not obvious that the worst-case joint remains feasible; if the paper only optimizes over the Fréchet class without enforcing the ball constraint on the joint, the reported shrinkage could be off. The choice of radius also looks like it will require case-by-case judgment rather than a data-driven default. This is aimed at econometricians and applied microeconomists who already work with heterogeneous treatment effects and worry about external validity. It is not paradigm-shifting, but the estimator is concrete enough that a careful referee could check the sharpness claim and the asymptotics in a few weeks. I would send it to peer review rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a distributionally robust estimator for predicting treatment effects under distribution shift, minimizing worst-case mean squared error over a Wasserstein ball around the source distribution. Due to the unidentified joint distribution of potential outcomes, it derives sharp upper and lower bounds on the minimax predictor by exploiting the Fréchet class consistent with the marginals of the potential outcomes. The resulting predictor preserves the sign of the source average treatment effect but shrinks it toward zero, with the degree of shrinkage depending on treatment effect heterogeneity. The paper establishes consistency and asymptotic normality of the bound estimators, proposes a two-step inference procedure, and discusses selection of the robustness radius.

Significance. If the central claims hold, the work offers a principled combination of distributionally robust optimization and partial identification to handle policy prediction across environments. The explicit shrinkage formula tied to heterogeneity provides an interpretable adjustment, and the asymptotic results support practical use. This could strengthen applications in econometrics where source and target populations differ, provided the Fréchet bounds remain sharp under the Wasserstein constraint on observables.

major comments (2)

[Abstract] Abstract: the claim that the Fréchet bounds are sharp for the minimax optimizer inside the Wasserstein ball is load-bearing for the shrinkage result. The ball constrains the law of the observed (X, Y) pair, yet the Fréchet class is defined solely on the marginals of the unidentified potential outcomes Y(0) and Y(1). No argument is given that the joint achieving the Fréchet extremum necessarily satisfies the Wasserstein distance constraint; if it lies outside the ball, the reported shrinkage factor is not guaranteed to be the true worst-case value.
[Abstract] Abstract / inference section: the two-step inference procedure for the bound estimators is described at a high level, but the paper does not detail how first-step estimation of the marginal distributions propagates into the asymptotic variance of the second-step bounds. This affects whether the claimed asymptotic normality holds uniformly over the robustness radius.

minor comments (1)

[Abstract] Abstract: the statement that the predictor 'preserves the sign of the average treatment effect under the source distribution' would benefit from an explicit statement of the conditions on the robustness radius under which this holds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to strengthen the relevant arguments and derivations.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the Fréchet bounds are sharp for the minimax optimizer inside the Wasserstein ball is load-bearing for the shrinkage result. The ball constrains the law of the observed (X, Y) pair, yet the Fréchet class is defined solely on the marginals of the unidentified potential outcomes Y(0) and Y(1). No argument is given that the joint achieving the Fréchet extremum necessarily satisfies the Wasserstein distance constraint; if it lies outside the ball, the reported shrinkage factor is not guaranteed to be the true worst-case value.

Authors: We agree that an explicit verification is needed to confirm that the Fréchet extremal joints remain feasible under the Wasserstein constraint on the observed data. In the revised manuscript we will add a supporting lemma showing that, because the Wasserstein ball is defined solely with respect to the observed marginal law of (X, Y) and the Fréchet extremals preserve the required marginals of Y(0) and Y(1) while respecting the mixture structure of the observed outcome, the extremal couplings lie inside the ball for all radii in the relevant range. This establishes sharpness of the bounds and validates the reported shrinkage formula. revision: yes
Referee: [Abstract] Abstract / inference section: the two-step inference procedure for the bound estimators is described at a high level, but the paper does not detail how first-step estimation of the marginal distributions propagates into the asymptotic variance of the second-step bounds. This affects whether the claimed asymptotic normality holds uniformly over the robustness radius.

Authors: We acknowledge that the current exposition of the two-step procedure is too concise. In the revision we will expand the inference section to derive the explicit influence function for the bound estimators, incorporating the first-step estimation error of the marginal distributions through the functional delta method. We will also establish uniform asymptotic normality over a compact interval of robustness radii under standard regularity conditions, and we will add a brief proof sketch together with any additional assumptions required for uniformity. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard Fréchet partial identification on marginals plus Wasserstein ball without self-referential reduction

full rationale

The paper's central derivation characterizes the minimax predictor via sharp bounds on the worst-case MSE inside the Wasserstein neighborhood, obtained by intersecting the Wasserstein ball (on observables) with the Fréchet class (on marginals of potential outcomes). This step relies on external mathematical objects (Fréchet-Hoeffding bounds and Wasserstein distance) rather than defining any quantity in terms of itself or renaming a fitted parameter as a prediction. The resulting shrinkage toward zero is an output of the optimization over the identified set and depends on observed heterogeneity; it is not presupposed by construction. No self-citations appear in the load-bearing steps, no uniqueness theorems are imported from the authors' prior work, and no ansatz is smuggled via citation. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Wasserstein neighborhood for modeling distributional shifts and on the Fréchet class providing sharp bounds for the unidentified joint distribution of potential outcomes; the robustness radius is a user-chosen free parameter.

free parameters (1)

robustness radius (Wasserstein ball size)
Controls the size of the distributional neighborhood and thereby the degree of shrinkage; chosen by the analyst rather than derived from data.

axioms (1)

domain assumption The joint distribution of potential outcomes is unidentified from the observed marginal distributions
Invoked to justify the use of the Fréchet class for partial identification of the worst-case error.

pith-pipeline@v0.9.0 · 5438 in / 1320 out tokens · 132972 ms · 2026-05-16T22:26:20.404066+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

minimize the worst-case mean squared error for the prediction of treatment effect over a class of distributions defined by a Wasserstein neighborhood... sharp upper and lower bounds... by exploiting the Fréchet class... preserves the sign... but is shrunk toward zero
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Vp = VU(P1,P0) = sup V(C(P1,P0)), Vo = VL(P1,P0) = inf V... Fréchet-Hoeffding inequality... Cov_CL ≤ Cov_P ≤ Cov_CU

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Moreover, we have Cov( ˙τ1, ˙σ 2

= 0 . Moreover, we have Cov( ˙τ1, ˙σ 2

work page
[2]

= E[(Y (1) − τ1)3] e , Cov( ˙τ0, ˙σ 2

work page
[3]

= E[(Y (0) − τ0)3] 1 − e , Var( ˙σ 2

work page
[4]

= E[((Y (1) − τ1)2 − σ 2 1)2] e , Var( ˙σ 2

work page
[5]

= E[((Y (0) − τ0)2 − σ 2 0)2] 1 − e . Step 1. Root- n negligibility of the remainder terms. Write ˆτ1 = ¯Y1 = ∑n i=1 TiYi ∑n i=1 Ti . Let n1 = ∑n i=1 Ti and ˆe = n1/n . Then: ˆτ1 = 1 n1 n∑ i=1 TiYi = 1 ˆe · 1 n n∑ i=1 TiYi = 1 ˆe [ 1 n n∑ i=1 Ti(Yi − τ1) + ˆeτ1 ] = τ1 + 1 ˆe · 1 n n∑ i=1 Ti(Yi − τ1) = τ1 + (1 e + Op(n− 1/ 2) ) · 1 n n∑ i=1 Ti(Yi − τ1) = τ...

work page
[6]

− 2Cov( ˙τ1, ˙θo) + 2Cov( ˙τ0, ˙θo) + 2Cov( ˙τ1, ˙γ) − 2Cov( ˙τ0, ˙γ) = 1 e E [ (Y (1) − τ1)3] − 1 1 − e E [ (Y (0) − τ0)3] − 2Cov( ˙τ1, ˙θo) + 2Cov( ˙τ0, ˙θo) + 2τ0Var( ˙τ1) − 2τ1Var( ˙τ0) Since inﬂuence functions have an expectation of zero, the covaria nce is just the expectation of their product: Cov( ˙τ1, ˙Q1(u)) = E[ ˙τ1 · ˙Q1(u)] = − 1 e2f1(Q1(u)) ...

work page
[7]

The terms in the ﬁrst three lines have been analyzed

+ 4τ2 0 Var( ˙τ1) + 4τ2 1 Var( ˙τ0) + 2τ0Cov( ˙σ 2 1, ˙τ1) + 2τ1Cov( ˙σ 2 0, ˙τ0) − 4τ0Cov( ˙τ1, ˙θo) − 4τ1Cov( ˙τ0, ˙θo) − 4τ0Cov( ˙τ1, ˙θp) − 4τ1Cov( ˙τ0, ˙θp) − 2Cov( ˙σ 2 1, ˙θo) − 2Cov( ˙σ 2 0, ˙θo) − 2Cov( ˙σ 2 1, ˙θp) − 2Cov( ˙σ 2 0, ˙θp) + 4Cov( ˙θo, ˙θp). The terms in the ﬁrst three lines have been analyzed. We will mainly foc us on the last two ...

work page 2005
[8]

The sequence {Cn}∞ n=1 is stochastically bounded uniformly in P ∈ P, i.e., for every ǫ > 0, there exists Mǫ < ∞ such that sup P ∈P P (|Cn|> M ǫ) < ǫ for all n ∈ N

work page
[9]

− − →0, i.e., for every δ > 0, lim n→∞ sup P ∈P P (|Dn|> δ) = 0

The sequence {Dn}∞ n=1 converges to zero in probability uniformly over P, denoted Dn u.p. − − →0, i.e., for every δ > 0, lim n→∞ sup P ∈P P (|Dn|> δ) = 0 . Then the product sequence {CnDn}∞ n=1 also converges to zero uniformly in probability over P: CnDn u.p. − − →0. Proof. Let ǫ > 0 and η > 0 be arbitrary. We need to show that there exists 70 N ∈ N such ...

work page
[10]

Then by Chebyshev’s inequal- ity: sup P ∈P P (|ˆe − e|> ǫ) ≤ sup P ∈P VarP (ˆe) ǫ2 ≤ C1 nǫ2 → 0 73 as n → ∞ , which gives ˆe u.p

(D.5) Since Ti ∼ Bernoulli(e) and by Assumption D.2, e ∈ [emin, e max] uniformly over P, we have: VarP (ˆe) = e(1 − e) n ≤ emax(1 − emin) n ≤ C1 n for some constant C1 < ∞ independent of P . Then by Chebyshev’s inequal- ity: sup P ∈P P (|ˆe − e|> ǫ) ≤ sup P ∈P VarP (ˆe) ǫ2 ≤ C1 nǫ2 → 0 73 as n → ∞ , which gives ˆe u.p. − − →e. The analysis of ˆτ0 and ˆτ1 ...

work page
[11]

The variance of G1 n is: VarP (G1 n) = Var P ( 1 n n∑ i=1 Zi ) = 1 n VarP (Zi), since the Zi are i.i.d

Under the standard assumptions, EP [Zi] = σ 2 1, so EP [G1 n] = 0. The variance of G1 n is: VarP (G1 n) = Var P ( 1 n n∑ i=1 Zi ) = 1 n VarP (Zi), since the Zi are i.i.d. across i. We now bound sup P ∈P VarP (Zi). Note that: EP [Z 2 i ] = EP [ (Ti(Yi − τ1)2 e )2] = 1 e2 EP [ Ti(Yi − τ1)4] = 1 e EP [ (Y (1) − τ1)4] , where we use that Ti is independent of ...

work page
[12]

− − →σ 2 0

(D.7) A similar argument gives ˆσ 2 0 u.p. − − →σ 2 0. 76 Step 2: Uniform consistency of fourth central moments. Deﬁne: ˆµ 4, 1 = 1 n1 n∑ i=1 Ti(Yi − ˆτ1)4, ˆµ 4, 0 = 1 n0 n∑ i=1 (1 − Ti)(Yi − ˆτ0)4, where n1 = ∑n i=1 Ti and n0 = n − n1. As shown in Lemma D.2 and the previous step, it is suﬃcient to prove the result with n1 = nˆe replaced by ne. Then we o...

work page
[13]

According to the continuous mapping theorem, when applied uniform ly: sup P ∈P |ˆσ 2 τ − σ 2 τ | P − → 0

= σ 2 1 e + σ 2 0 1− e is contin- uous and bounded on the compact set {e ∈ [emin, e max], σ 2 1, σ 2 0 ∈ [σ 2 min, σ 2 max]}. According to the continuous mapping theorem, when applied uniform ly: sup P ∈P |ˆσ 2 τ − σ 2 τ | P − → 0. (D.8) Step 4: Uniform consistency of σ 2 +. The estimator is: ˆσ 2 + = ( 1 + ˆσ 2 0 ˆσ 2 1 )2 ˆµ 4, 1 − (ˆσ 2 1)2 ˆe − ( 1 + ...

work page
[14]

By Assumption D.4 (uniform separation), these coeﬃcients are uniformly bounded aw ay from zero, ensuring continuity

with (1 − σ 2 0/σ 2 1), etc. By Assumption D.4 (uniform separation), these coeﬃcients are uniformly bounded aw ay from zero, ensuring continuity. Thus: sup P ∈P |ˆσ 2 − − σ 2 − | P − → 0. (D.10) Step 6: Uniform consistency of oﬀ-diagonal elements. The uni- form consistency of the oﬀ-diagonal elements can be obtained by t he Cauchy- Schwarz inequality and ...

work page 2019

[1] [1]

Moreover, we have Cov( ˙τ1, ˙σ 2

= 0 . Moreover, we have Cov( ˙τ1, ˙σ 2

work page

[2] [2]

= E[(Y (1) − τ1)3] e , Cov( ˙τ0, ˙σ 2

work page

[3] [3]

= E[(Y (0) − τ0)3] 1 − e , Var( ˙σ 2

work page

[4] [4]

= E[((Y (1) − τ1)2 − σ 2 1)2] e , Var( ˙σ 2

work page

[5] [5]

= E[((Y (0) − τ0)2 − σ 2 0)2] 1 − e . Step 1. Root- n negligibility of the remainder terms. Write ˆτ1 = ¯Y1 = ∑n i=1 TiYi ∑n i=1 Ti . Let n1 = ∑n i=1 Ti and ˆe = n1/n . Then: ˆτ1 = 1 n1 n∑ i=1 TiYi = 1 ˆe · 1 n n∑ i=1 TiYi = 1 ˆe [ 1 n n∑ i=1 Ti(Yi − τ1) + ˆeτ1 ] = τ1 + 1 ˆe · 1 n n∑ i=1 Ti(Yi − τ1) = τ1 + (1 e + Op(n− 1/ 2) ) · 1 n n∑ i=1 Ti(Yi − τ1) = τ...

work page

[6] [6]

− 2Cov( ˙τ1, ˙θo) + 2Cov( ˙τ0, ˙θo) + 2Cov( ˙τ1, ˙γ) − 2Cov( ˙τ0, ˙γ) = 1 e E [ (Y (1) − τ1)3] − 1 1 − e E [ (Y (0) − τ0)3] − 2Cov( ˙τ1, ˙θo) + 2Cov( ˙τ0, ˙θo) + 2τ0Var( ˙τ1) − 2τ1Var( ˙τ0) Since inﬂuence functions have an expectation of zero, the covaria nce is just the expectation of their product: Cov( ˙τ1, ˙Q1(u)) = E[ ˙τ1 · ˙Q1(u)] = − 1 e2f1(Q1(u)) ...

work page

[7] [7]

The terms in the ﬁrst three lines have been analyzed

+ 4τ2 0 Var( ˙τ1) + 4τ2 1 Var( ˙τ0) + 2τ0Cov( ˙σ 2 1, ˙τ1) + 2τ1Cov( ˙σ 2 0, ˙τ0) − 4τ0Cov( ˙τ1, ˙θo) − 4τ1Cov( ˙τ0, ˙θo) − 4τ0Cov( ˙τ1, ˙θp) − 4τ1Cov( ˙τ0, ˙θp) − 2Cov( ˙σ 2 1, ˙θo) − 2Cov( ˙σ 2 0, ˙θo) − 2Cov( ˙σ 2 1, ˙θp) − 2Cov( ˙σ 2 0, ˙θp) + 4Cov( ˙θo, ˙θp). The terms in the ﬁrst three lines have been analyzed. We will mainly foc us on the last two ...

work page 2005

[8] [8]

The sequence {Cn}∞ n=1 is stochastically bounded uniformly in P ∈ P, i.e., for every ǫ > 0, there exists Mǫ < ∞ such that sup P ∈P P (|Cn|> M ǫ) < ǫ for all n ∈ N

work page

[9] [9]

− − →0, i.e., for every δ > 0, lim n→∞ sup P ∈P P (|Dn|> δ) = 0

The sequence {Dn}∞ n=1 converges to zero in probability uniformly over P, denoted Dn u.p. − − →0, i.e., for every δ > 0, lim n→∞ sup P ∈P P (|Dn|> δ) = 0 . Then the product sequence {CnDn}∞ n=1 also converges to zero uniformly in probability over P: CnDn u.p. − − →0. Proof. Let ǫ > 0 and η > 0 be arbitrary. We need to show that there exists 70 N ∈ N such ...

work page

[10] [10]

Then by Chebyshev’s inequal- ity: sup P ∈P P (|ˆe − e|> ǫ) ≤ sup P ∈P VarP (ˆe) ǫ2 ≤ C1 nǫ2 → 0 73 as n → ∞ , which gives ˆe u.p

(D.5) Since Ti ∼ Bernoulli(e) and by Assumption D.2, e ∈ [emin, e max] uniformly over P, we have: VarP (ˆe) = e(1 − e) n ≤ emax(1 − emin) n ≤ C1 n for some constant C1 < ∞ independent of P . Then by Chebyshev’s inequal- ity: sup P ∈P P (|ˆe − e|> ǫ) ≤ sup P ∈P VarP (ˆe) ǫ2 ≤ C1 nǫ2 → 0 73 as n → ∞ , which gives ˆe u.p. − − →e. The analysis of ˆτ0 and ˆτ1 ...

work page

[11] [11]

The variance of G1 n is: VarP (G1 n) = Var P ( 1 n n∑ i=1 Zi ) = 1 n VarP (Zi), since the Zi are i.i.d

Under the standard assumptions, EP [Zi] = σ 2 1, so EP [G1 n] = 0. The variance of G1 n is: VarP (G1 n) = Var P ( 1 n n∑ i=1 Zi ) = 1 n VarP (Zi), since the Zi are i.i.d. across i. We now bound sup P ∈P VarP (Zi). Note that: EP [Z 2 i ] = EP [ (Ti(Yi − τ1)2 e )2] = 1 e2 EP [ Ti(Yi − τ1)4] = 1 e EP [ (Y (1) − τ1)4] , where we use that Ti is independent of ...

work page

[12] [12]

− − →σ 2 0

(D.7) A similar argument gives ˆσ 2 0 u.p. − − →σ 2 0. 76 Step 2: Uniform consistency of fourth central moments. Deﬁne: ˆµ 4, 1 = 1 n1 n∑ i=1 Ti(Yi − ˆτ1)4, ˆµ 4, 0 = 1 n0 n∑ i=1 (1 − Ti)(Yi − ˆτ0)4, where n1 = ∑n i=1 Ti and n0 = n − n1. As shown in Lemma D.2 and the previous step, it is suﬃcient to prove the result with n1 = nˆe replaced by ne. Then we o...

work page

[13] [13]

According to the continuous mapping theorem, when applied uniform ly: sup P ∈P |ˆσ 2 τ − σ 2 τ | P − → 0

= σ 2 1 e + σ 2 0 1− e is contin- uous and bounded on the compact set {e ∈ [emin, e max], σ 2 1, σ 2 0 ∈ [σ 2 min, σ 2 max]}. According to the continuous mapping theorem, when applied uniform ly: sup P ∈P |ˆσ 2 τ − σ 2 τ | P − → 0. (D.8) Step 4: Uniform consistency of σ 2 +. The estimator is: ˆσ 2 + = ( 1 + ˆσ 2 0 ˆσ 2 1 )2 ˆµ 4, 1 − (ˆσ 2 1)2 ˆe − ( 1 + ...

work page

[14] [14]

By Assumption D.4 (uniform separation), these coeﬃcients are uniformly bounded aw ay from zero, ensuring continuity

with (1 − σ 2 0/σ 2 1), etc. By Assumption D.4 (uniform separation), these coeﬃcients are uniformly bounded aw ay from zero, ensuring continuity. Thus: sup P ∈P |ˆσ 2 − − σ 2 − | P − → 0. (D.10) Step 6: Uniform consistency of oﬀ-diagonal elements. The uni- form consistency of the oﬀ-diagonal elements can be obtained by t he Cauchy- Schwarz inequality and ...

work page 2019