Semi-supervised inference for treatment heterogeneity

Yilizhati Anniwaer; Yuqian Zhang

REVIEW 2 major objections 5 minor 12 references

Semi-supervised inference for treatment heterogeneity

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that large pools of unlabeled covariate and treatment records can be safely used to estimate treatment-effect heterogeneity, with asymptotic variance never worse than supervised methods.

desk verdict Solid semi-supervised estimators for treatment heterogeneity; the optimal-weighting trick for ETH is the real contribution, and it holds up, though the real-data demo is narrower than the abstract claims. read the letter →

arxiv 2509.05048 v1 pith:57GIP3V6 submitted 2025-09-05 stat.ME

Yilizhati Anniwaer , Yuqian Zhang This is my paper

classification stat.ME

keywords semi-supervisedlearningtreatmentheterogeneitytotalexplainedconditionalaverageeffectdoublyrobustestimationhigh-dimensionalstatisticsoptimalweighting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Treatment-effect heterogeneity tells doctors whether a one-size-fits-all treatment is enough or whether decisions should depend on patient characteristics. The paper develops semi-supervised estimators for two heterogeneity measures—the total treatment heterogeneity, Var[τ(X)], and the explained treatment heterogeneity, Var[Wᵀβ*], of a working linear model—that use a large unlabeled pool of covariate/treatment records alongside a smaller labeled set with outcomes. For total heterogeneity it proves asymptotic normality under weaker model-correctness and sparsity conditions than supervised estimators, with an asymptotic variance reduced by (1−n/N)Var[(τ(X)−Eτ)²]. For explained heterogeneity it shows that naively pooling unlabeled data can lose efficiency, and fixes this with a data-dependent reweighting of labeled and unlabeled samples that achieves variance no larger than the supervised estimator. If correct, the result means cheap unlabeled records can be recycled to sharpen personalized-treatment statistics without sacrificing validity.

What carries the argument

The central object is the reweighted plug-in-plus-debiasing estimator for ETH: θ̂_RW = ω_L Σ_labeled (D̂ᵀβ̂)² + ω_U Σ_unlabeled (D̂ᵀβ̂)² + (2/n) Σ_labeled D̂ᵀβ̂ ε̂, with weights constrained by nω_L + mω_U = 1 to avoid bias. Minimizing the asymptotic variance over this constraint gives ω_L* = (nB − mC)/(nNB) and ω_U* = (B + C)/(NB), where B = Var[(Dᵀβ*)²] and C = Cov[2εDᵀβ*, (Dᵀβ*)²]. The sign of C decides whether labeled or unlabeled records should receive extra weight. For TTH, the analogous mechanism is a cross-fitted doubly robust pseudo-outcome plug-in with an additional labeled-only debiasing term; the unlabeled pool mainly improves the plug-in component and accelerates propensity-score

What would settle it

Run the semi-supervised TTH estimator under the paper's Model 2 misspecification with n = 1000 and m = 8000 and check 95% confidence interval coverage across many repetitions: if coverage does not approach 95% or the empirical variance is not within Monte Carlo error of Var[2ξh(X)] + (n/N)Var[h²(X)], the central guarantee fails. For ETH, estimate B and C and compare the variance of the optimally weighted estimator with the supervised estimator; any clear violation of σ_OW² ≤ σ_sup² would falsify Theorem 4.

Watch

Extended reading notes

Core claim

The central claim is that both TTH and ETH can be estimated semi-supervisedly with provable efficiency guarantees. Theorem 1 gives √n(θ̂_TTH − θ_TTH)/σ_TTH ⇒ N(0,1) with σ_TTH² = Var[2ξh(X)] + (n/N)Var[h²(X)], which is at most the supervised variance Var[2ξh(X)] + Var[h²(X)]; this holds even when the outcome model is misspecified, provided the propensity score, estimated from the pooled labeled and unlabeled data, has squared L2 error o_p(n⁻¹). Theorem 4 gives the corresponding ETH result: the optimally weighted estimator has variance σ_OW² = A + nB/N + 2nC/N − mC²/(NB), no larger than the direct semi-supervised variance A + (n/N)B + 2(n/N)C and the supervised variance A + B + 2C. The method

Load-bearing premise

The entire robustness gain over supervised methods rests on the unlabeled data containing treatment assignments and being large enough that the propensity score can be estimated from the pooled data with squared error shrinking faster than 1/n; if only covariates are unlabeled, the robustness advantage disappears.

Editorial extensions

If this is right

Semi-supervised TTH estimation remains asymptotically normal when the outcome regression is misspecified, as long as the propensity score is estimated from a large enough pooled sample; supervised TTH estimators require both models to be correct.
The TTH variance reduction is (1−n/N)Var[(τ(X)−Eτ)²], so the gain disappears only when unlabeled records are negligible or the CATE is nearly constant.
For ETH, the optimally weighted estimator dominates the direct equal-weight semi-supervised estimator by mC²/(NB) and the supervised estimator by m(B+C)²/(NB), so adding unlabeled data is never an efficiency loss.
The sign of C tells practitioners whether to up-weight labeled or unlabeled records; when the sub-CATE model is correctly specified, C = 0 and equal weights 1/N are optimal.
When unlabeled records contain only covariates, efficiency gains remain but the model-robustness benefit disappears, because the propensity score must then be estimated from labeled data alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same reweighting logic likely extends to other functionals with a plug-in-plus-debiasing structure, such as semi-supervised U-statistics; the covariance C plays the role of a direction-of-harm that determines when unlabeled data should be down-weighted.
In multi-arm randomized trials, treating non-target arms as unlabeled—as the ACTG175 analysis does—can be generalized into a design principle: randomization makes missing-completely-at-random automatic, so every arm can contribute covariate information to pairwise heterogeneity estimates.
The safety guarantee is tied to correct propensity-score specification; an observational-study user should therefore spend model-selection effort on the propensity score, since outcome-model errors are forgiven by large unlabeled pools only when the propensity score converges fast enough.
If unlabeled data contain only covariates, the robustness claim lapses; a plausible testable variant is whether a correctly specified propensity score estimated from labeled data at n^{-1/2} still preserves the ETH variance guarantee, possibly with a modified constant.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid methodological paper, and the central claim survives reading. The genuinely new piece is the re-weighting scheme for the explained treatment heterogeneity estimator, which provably makes the semi-supervised variance no larger than the supervised one even when outcome models are misspecified. The TTH estimator is a cleaner adaptation of existing debiased semi-supervised means to DR pseudo-outcomes, but the ETH estimator and its variance formula (Theorem 4) are the reason to read it. The algebra is consistent and the proofs are detailed in the supplement. Simulations line up with the theory, including the expected dip for the parametric estimator when nuisance models are misspecified but the semi-parametric version recovers.

Soft spots: no code is shipped, so the practical side — especially estimating the weights A, B, C with B bounded away from zero — is not stress-tested. The rate conditions in Assumptions 3 and 4 are real: they require either product rates or a propensity error o_p(n^{-1/2}) in L2 from pooled data, which needs unlabeled treatment assignments. That is exactly where the ACTG175 example overreaches. The unlabeled arms are patients on other treatments; they don't have the target A. So the real-data application demonstrates efficiency gains from extra covariates only, not the robustness guarantee. That matches Remark 2, but it contradicts the abstract's unqualified robustness claim. It's a scope issue, not a flaw in the theorems. Also the paper doesn't handle weight estimation in the simulations when B is small; a minor gap.

Citation pattern is reasonable. The self-citations are to tools they actually build on, and the new-result claim is honest. I'd send this to a serious statistical methodology referee. It belongs in a good journal after a revision that either adjusts the abstract and real-data language or extends the application to a setting with unlabeled treatment indicators.

Referee Report

2 major / 5 minor

Summary. The paper develops semi-supervised estimators for two treatment-effect heterogeneity measures: the total treatment heterogeneity (TTH), Var[τ(X)], and the explained treatment heterogeneity (ETH) of a linear working model, Var[W^Tβ*]. The labeled sample contains (X,A,Y); the unlabeled sample is assumed in the main theory to contain (X,A). For TTH, a debiased plug-in estimator is constructed from doubly robust pseudo-outcomes with cross-fitting; Theorem 1 gives √n-asymptotic normality with variance Var(2ξh(X)) + (n/N)Var(h^2(X)). For ETH, the paper shows that a direct semi-supervised estimator can have larger asymptotic variance than a supervised estimator, and then introduces an optimally weighted estimator whose weights minimize the leading variance term. Theorem 4 proves that the resulting variance is no larger than either the direct semi-supervised or the supervised variance, and that the estimated weights do not affect the first-order distribution. High-dimensional Lasso rates and a semi-parametric extension are provided, followed by simulations and an ACTG175 application.

Significance. The paper makes a genuine methodological contribution. The optimal-weighting construction is clean: the weights are derived by minimizing an explicit quadratic variance formula, and the claimed dominance over both direct semi-supervised and supervised estimators follows algebraically. The supplement contains detailed proofs, and the simulation results are broadly consistent with the stated variance formulas. The main limitation is scope. The robustness claim—outcome regression may be misspecified as long as the propensity score is estimated from the pooled data—requires the unlabeled sample to contain the target binary treatment indicator. The ACTG application does not satisfy this requirement as presented, so the real-data example does not demonstrate the headline robustness gain. This is a fixable but load-bearing qualification, not a defect in the core proofs.

major comments (2)

[Section 2.2, Assumption 3(b), Remark 2] The headline robustness claim is conditional on the unlabeled data containing the target treatment indicator A. Assumption 3(b) requires EX|\hatπ(X)-π(X)|^2 = o_p(n^{-1}) from the pooled labeled and unlabeled sample. This is only possible when unlabeled units have the same binary treatment assignment A whose propensity is being estimated. Remark 2 concedes that when unlabeled data contain only covariates, the robustness improvement disappears. This restriction is not reflected in the abstract or the introduction, where improved robustness is presented as a general benefit. The same issue also affects the ETH results in Section 3: Theorems 3 and 4 condition (b), nsγ log d = o(N), relies on π being estimated from all N units; with covariate-only unlabeled data the propensity error would be governed by n rather than N, and the stated robustness gain is lost. The scope of the main claims sho
[Section 4.2, ACTG175 application] The application does not satisfy the semi-supervised assumption used in the main theory. The unlabeled units are patients randomized to the two non-target treatment arms. These units do not have a value of the target binary treatment A for the comparison ZDV+ddI versus ddI, so Algorithm 1's Step 6, which estimates the target propensity score from G = labeled ∪ unlabeled, is not well-defined for them. The joint distribution of (X,A) is not the same in the labeled and unlabeled groups unless one redefines the problem. If the authors rely on randomization to treat the propensity score as known, they should say so explicitly and present the analysis as a covariate-only semi-supervised efficiency gain, which per Remark 2 does not include robustness to outcome misspecification. As written, Table 5 cannot be used to support the claim that the proposed semi-supervised method improves robustness

minor comments (5)

[Section 4.2, Table 5] The p-values appear to be one-sided for H0: θ=0 versus H1: θ>0, while the confidence intervals are two-sided. This should be stated explicitly; otherwise the p=0.051 with a CI crossing zero may confuse readers.
[Tables 1 and 2] Tables 1 and 2 combine TTH and ETH results because the true CATE is linear with W=X. A brief sentence in the text makes this clear, but it would help to repeat it in the table captions.
[Equation (13) and Algorithm 3] The notation for the estimated weights is slightly ambiguous: (13) writes \hatω_L^{(k)} and \hatω_U^{(k)} inside sums over k, but the surrounding text also refers to 'the weights'. Clarify that the weights are fold-specific or that the displayed expression is a sum over folds with fold-specific weights.
[Section 5, Discussion] The statement that the framework 'can be readily extended to non-linear or even non-parametric working models' is speculative and not supported by the current proofs. It would be appropriate to label this as a conjecture or future work.
[Throughout] There are occasional notation ambiguities, e.g., the use of ¢/BD/CA symbols in equations (5)-(7) from the original LaTeX; the final journal version should ensure these are typeset correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variance reductions are analytic consequences of the estimator expansions, not fitted inputs or self-referential conclusions.

full rationale

I walked the derivation chain for both main claims. The TTH result (Theorem 1, Remark 1) derives σ²_TTH = Var[2ξh(X)] + (n/N)Var[h²(X)] from an influence-function expansion of the proposed cross-fitted estimator, and the comparison with the supervised variance is algebra: the supervised variance is the same expression with n/N replaced by 1. The variance reduction (1 − n/N)Var[h²(X)] is a mathematical consequence of averaging the h² term over N rather than n observations, not a fitted result. For ETH, the optimally weighted estimator (Theorem 4, Remark 4) is obtained by minimizing the explicit quadratic variance formula σ²_RW(ω_L, ω_U) subject to nω_L + mω_U = 1. The population weights are the minimizers of that quadratic, and the inequalities σ²_OW ≤ σ²_para and σ²_OW ≤ σ²_sup follow by direct algebra (mC²/(NB) ≥ 0 and m(B+C)²/(NB) ≥ 0). The data-dependent weights are shown in Lemma 10 to converge to these population optimizers fast enough not to affect the first-order variance; this is a standard plug-in/estimation step, not a circular fit. The self-citations to Zhang & Bradic (2022) and Zhang et al. (2023) are used for background estimators and technical lemmas (e.g., semi-supervised mean debiasing, restricted eigenvalue bounds); they are not used to import the paper's central efficiency or robustness conclusions. The paper's robustness claim does depend on Assumption 3(b), and Remark 2 explicitly concedes that covariate-only unlabeled data lose the robustness improvement; this is a substantive assumption/limitation, but it is not an equivalence between the input and the claimed output. No step in the paper reduces, by construction, to its own inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or entities are introduced; the optimal weights are functions of existing data distributions. The free parameters are standard Lasso tuning parameters. All axioms are either standard causal assumptions or technical rate conditions used in the semi-supervised doubly robust literature.

free parameters (3)

λ_α (Lasso penalty for outcome regression) = cross-validated in simulation; theory requires λ_α ≍ sqrt(log(d)/n)
Standard regularization parameter in eq. (5), not an ad hoc device.
λ_γ (Lasso penalty for propensity score) = cross-validated in simulation; theory requires λ_γ ≍ sqrt(log(d)/N)
Standard regularization parameter in eq. (6), uses total sample size N.
λ_β (Lasso penalty for CATE working model) = cross-validated in simulation; theory requires λ_β ≍ sqrt(log(p)/n)
Standard regularization parameter in eq. (7), uses labeled size n.

assumptions (5)

domain assumption Assumption 1: unconfoundedness, consistency, overlap
Standard causal identification conditions invoked in Section 2.1 to define τ(x) and TTH.
domain assumption MCAR labeling: labeled and unlabeled (X,A) share the same joint distribution
Assumed in Section 2; needed for the semi-supervised gains and for treating non-target trial arms as unlabeled.
domain assumption Assumption 3: nuisance rate conditions (product-rate or π-rate o_p(n^{-1}))
Standard in doubly robust inference; enables outcome model misspecification when propensity score is well estimated.
domain assumption Assumption 4: Var_X[hat τ - τ] = o_p(n^{-1/2})
Strong rate condition on the CATE estimator used in Theorem 1 for TTH asymptotic normality.
domain assumption Assumptions 5-7: sub-Gaussian errors, sub-Gaussian covariates, overlap, sparsity
Technical conditions for high-dimensional Lasso results in Theorems 2-4.

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Cite this review

Pith. "Pith review of Semi-supervised inference for treatment heterogeneity." pith.science (2026). https://pith.science/paper/57GIP3V6

@misc{pith2026250905048,
  author       = {Pith},
  title        = {Pith review of: Semi-supervised inference for treatment heterogeneity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/57GIP3V6}},
  note         = {Machine review of arXiv:2509.05048}
}

read the original abstract

In causal inference, measuring treatment heterogeneity is crucial as it provides scientific insights into how treatments influence outcomes and guides personalized decision-making. In this work, we study semi-supervised settings where a labeled dataset is accompanied by a large unlabeled dataset, and develop semi-supervised estimators for two measures of treatment heterogeneity: the total treatment heterogeneity (TTH) and the explained treatment heterogeneity (ETH) of a simplified working model. We propose semi-supervised estimators for both quantities and demonstrate their improved robustness and efficiency compared with supervised methods. For ETH estimation, we show that direct semi-supervised approaches may result in efficiency loss relative to supervised counterparts. To address this, we introduce a re-weighting strategy that assigns data-dependent weights to labeled and unlabeled samples to optimize efficiency. The proposed approach guarantees an asymptotic variance no larger than that of the supervised method, ensuring its safe use. We evaluate the performance of the proposed estimators through simulation studies and a real-data application based on an AIDS clinical trial.

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Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Deﬁne ˆθ(k) TTH := |Gk|−1 ∑ i∈Gk [ ˆh(−k)(Xi) ] 2 + 2˜n−1∑ i∈Ik ˆh(−k)(Xi) [ ˆϕ(−k)(Zi) − ˆτ − ˆh(−k)(Xi) ] , ˜θ(k) TTH := |Gk|−1 ∑ i∈Gk [ ˆτ (−k)(Xi) −ν(−k) ] 2 + 2˜n−1∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ][ ϕ∗(Zi) −τ − ˆτ (−k)(Xi) +ν(−k) ] , ˇθ(k) TTH := |Gk|−1 ∑ i∈Gk [h(Xi)]2 + 2˜n−1∑ i∈Ik h(Xi)ξi, where ˆh(−k)(Xi) = ˆτ (−k)(Xi)−|Gk|−1∑ i∈Gk ˆτ (−k)(Xi),ν(−k) =...

work page 2019
[2]

For any k ≤K, we have 1 ˜n ∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ][ ˆϕ(−k)(Zi) −ϕ∗(Zi) ] = 1 K − 1 ∑ k′̸=k 1 ˜n ∑ i∈Ik {( ˆτ (−k)(Xi) −ν(−k) )[ ˆϕ(−k,−k′)(Zi) −ϕ∗(Zi) ]} . For any k′ ̸=k, consider the representation 1 ˜n ∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ][ ˆϕ(−k,−k′)(Zi) −ϕ∗(Zi) ] =A1 +A2 +A3, 38 where A1 = 1 ˜n ∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ] { [Yi(1) −µ∗ 1(Xi)] [ Ai ˆπ(...

work page 2023
[3]

≤ 2EX ⏐ ⏐ ⏐ ⏐ ⏐ ( µ∗ 1(X) − ˆµ(−k,−k′) 1 (X) )[ A π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 + 2EX ⏐ ⏐ ⏐ ⏐ ⏐ ( µ∗ 0(X) − ˆµ(−k,−k′) 0 (X) )[ 1 −A 1 −π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 ≤ 2 √ EX ⏐ ⏐ ⏐ ( µ∗ 1(X) − ˆµ(−k,−k′) 1 (X) )⏐ ⏐ ⏐ 4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ A π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 4 + 2 √ EX ⏐ ⏐ ⏐ ( µ∗ 0(X) − ˆµ(−k,−k′) 0 (X) )⏐ ⏐ ⏐ 4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ 1 −A 1 −π∗(X) − 1 ] ⏐ ⏐ ⏐...

work page 2023
[4]

(S37) It follows that ( ˆQ(k) )2 −θ2 ETH = ( ˆQ(k) +θETH)( ˆQ(k) −θETH) (i) =Op(1)op(1) = op(1), (S38) where (i) holds from ( S36) and ( S37)

= O(1). (S37) It follows that ( ˆQ(k) )2 −θ2 ETH = ( ˆQ(k) +θETH)( ˆQ(k) −θETH) (i) =Op(1)op(1) = op(1), (S38) where (i) holds from ( S36) and ( S37). According to ( S29), ( S30), and ( S38), ˆB(k) =B +op(1). (S39) Now, we establish the consistency of ˆC (k). By Lemma 9, ˆβ(−k)⊤  E(W ) − |Gk|−1 ∑ i∈Gk Wi   1 ˜n ∑ i∈Ik 2ˆǫi (k) [ ( β∗⊤ Di )2 −θETH ] =o...

work page
[5]

Finally, ∥ǫ∥ψ2 = ∥ϕ∗(Z) − W ⊤β∗∥ψ2 ≤ ∥ϕ∗(Z)∥ψ2 + ∥ W ⊤β∗∥ψ2 (i) ≤O(1) +σx ∥ β∗∥ 2 =O(1), where (i) holds from ∥ϕ∗(Z)∥ψ2 =O(1) and Assumption 6

Hence, ∥ β∗∥ 2 =O(1). Finally, ∥ǫ∥ψ2 = ∥ϕ∗(Z) − W ⊤β∗∥ψ2 ≤ ∥ϕ∗(Z)∥ψ2 + ∥ W ⊤β∗∥ψ2 (i) ≤O(1) +σx ∥ β∗∥ 2 =O(1), where (i) holds from ∥ϕ∗(Z)∥ψ2 =O(1) and Assumption 6. 48 Proof of Lemma 7. By Jensen’s inequality, sup ∥ α∥ 2≤1 |E[W ⊤α]| ≤ sup∥ α∥ 2=1 E|W ⊤α| = O(1), where the last step holds by Assumption 6 and Lemma 1. Since any bounded variable is sub-Gaus...

work page
[6]

Additionally, 1 ˜n ∑ i∈Ik [ ˆϕ(−k)(Zi) −ϕ∗(Zi) ] 4 =op(1) (S45) 63 holds from Lemma 7 and Markov’s inequality. Therefore, 1 ˜n ∑ i∈Ik [ 2 ( ˆǫ(k) i −ǫi ) β∗⊤ Di ] 2 = 4 ˜n ∑ i∈Ik [ ˆϕ(−k)(Zi) −ϕ∗(Zi) +τ − ˆτpara − ˆβ(−k)⊤ ˆD(k) i + β∗⊤ Di ] 2( β∗⊤ Di )2 =O   4 ˜n ∑ i∈Ik {[ ˆϕ(−k)(Zi) −ϕ∗(Zi) ] 2 + (τ − ˆτpara)2 + ( β∗⊤ Di − ˆβ(−k)⊤ ˆD(k) i )2} ( β∗⊤ Di ...

work page
[7]

Since E(a −b)2 ≤ 2E(a2 +b2), we have EX(∆2

log(p) n ) . Since E(a −b)2 ≤ 2E(a2 +b2), we have EX(∆2

work page
[8]

Hence, EX          1 ∑ k′̸=k |Ik′| ∑ k′̸=k ∑ i∈Ik′ /BD Ai=a∆2iWi       2 ∞    =Op ( sγ log(d) log(p) nN ) =op ( log(p) n ) , since sγ log(d) = o(N)

≤ 2EX ⏐ ⏐ ⏐ ⏐ ⏐[Y (1) −µ∗ 1(X)] [ A ˆπ(−k,−k′)(X) − A π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 + 2EX ⏐ ⏐ ⏐ ⏐ ⏐[Y (0) −µ∗ 0(X)] [ 1 −A 1 − ˆπ(−k,−k′)(X) − 1 −A 1 −π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 ≤ 2 √ EX|A [Y (1) −µ∗ 1(X)] |4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ 1 ˆπ(−k,−k′)(X) − 1 π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 4 + 2 √ EX|(1 −A) [Y (0) −µ∗ 0(X)] |4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ 1 1 − ˆπ(−k,−k′)(X) − 1 1 −π∗(X) ] ⏐ ⏐ ⏐ ⏐ ...

work page

Show all 12 references

[9]

Besides, |Gk|−1 ∑ i∈Gk ( ˆβ(−k)⊤ Di )4 = |Gk|−1 ∑ i∈Gk [ β∗⊤ Di + ( ˆβ(−k) − β∗ )⊤ Di ] 4 = |Gk|−1 ∑ i∈Gk ( β∗⊤ Di )4 + 4|Gk|−1 ∑ i∈Gk ( β∗⊤ Di )3[ ( ˆβ(−k) − β∗ )⊤ Di ] + 6|Gk|−1 ∑ i∈Gk ( β∗⊤ Di )2[ ( ˆβ(−k) − β∗ )⊤ Di ] 2 + 4|Gk|−1 ∑ i∈Gk ( β∗⊤ Di )[ ( ˆβ(−k) − β∗ )⊤ Di ] 3 ...
[10]

≤ 2EX ⏐ ⏐ ⏐ ⏐ ⏐[Y (1) −µ∗ 1(X)] [ A ˆπ(−k,−k′)(X) − A π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 + 2EX ⏐ ⏐ ⏐ ⏐ ⏐[Y (0) −µ∗ 0(X)] [ 1 −A 1 − ˆπ(−k,−k′)(X) − 1 −A 1 −π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 =Op ( EX|ˆπ(−k,−k′)(X) −π(X)|2 ) and EX(∆2
[11]

Repeating the steps in the proof of Theorem 2, under Assumption 3, 3∑ l=1       1 ∑ k′̸=k |Ik′| ∑ k′̸=k ∑ i∈Ik′ ∆liWi       ∞ =Op   √ log(p) n + √ EX(∆2

≤ 2EX ⏐ ⏐ ⏐ ⏐ ⏐ ( µ∗ 1(X) − ˆµ(−k,−k′) 1 (X) )[ A π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 + 2EX ⏐ ⏐ ⏐ ⏐ ⏐ ( µ∗ 0(X) − ˆµ(−k,−k′) 0 (X) )[ 1 −A 1 −π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 =Op ( max a∈{0,1} EX|ˆµ(−k,−k′) a (X) −µ∗ a(X)|2 ) . Repeating the steps in the proof of Theorem 2, under Assumption 3, 3∑ ...
[12]

This ensures the upper bound ( S61)

log(p) n   =Op   √ log(p) n  . This ensures the upper bound ( S61). Moreover, by ( S62) and ( S63), we also have EX(∆2 4) (i) =op(n−1), EX(∆2 5) (ii) = op(n−1), ∆6 (iii) = 0 , where (i) holds when EX|ˆπ(−k,−k′)(X) −π(X)|2|ˆµ(−k,−k′) a (X) −µa(X)|2 =op(n−1) for each a ∈ {...

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Reviewed August 5, 2026 · model on record in the stance chip above.

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Deﬁne ˆθ(k) TTH := |Gk|−1 ∑ i∈Gk [ ˆh(−k)(Xi) ] 2 + 2˜n−1∑ i∈Ik ˆh(−k)(Xi) [ ˆϕ(−k)(Zi) − ˆτ − ˆh(−k)(Xi) ] , ˜θ(k) TTH := |Gk|−1 ∑ i∈Gk [ ˆτ (−k)(Xi) −ν(−k) ] 2 + 2˜n−1∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ][ ϕ∗(Zi) −τ − ˆτ (−k)(Xi) +ν(−k) ] , ˇθ(k) TTH := |Gk|−1 ∑ i∈Gk [h(Xi)]2 + 2˜n−1∑ i∈Ik h(Xi)ξi, where ˆh(−k)(Xi) = ˆτ (−k)(Xi)−|Gk|−1∑ i∈Gk ˆτ (−k)(Xi),ν(−k) =...

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For any k ≤K, we have 1 ˜n ∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ][ ˆϕ(−k)(Zi) −ϕ∗(Zi) ] = 1 K − 1 ∑ k′̸=k 1 ˜n ∑ i∈Ik {( ˆτ (−k)(Xi) −ν(−k) )[ ˆϕ(−k,−k′)(Zi) −ϕ∗(Zi) ]} . For any k′ ̸=k, consider the representation 1 ˜n ∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ][ ˆϕ(−k,−k′)(Zi) −ϕ∗(Zi) ] =A1 +A2 +A3, 38 where A1 = 1 ˜n ∑ i∈Ik [ ˆτ (−k)(Xi) −ν(−k) ] { [Yi(1) −µ∗ 1(Xi)] [ Ai ˆπ(...

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≤ 2EX ⏐ ⏐ ⏐ ⏐ ⏐ ( µ∗ 1(X) − ˆµ(−k,−k′) 1 (X) )[ A π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 + 2EX ⏐ ⏐ ⏐ ⏐ ⏐ ( µ∗ 0(X) − ˆµ(−k,−k′) 0 (X) )[ 1 −A 1 −π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 ≤ 2 √ EX ⏐ ⏐ ⏐ ( µ∗ 1(X) − ˆµ(−k,−k′) 1 (X) )⏐ ⏐ ⏐ 4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ A π∗(X) − 1 ] ⏐ ⏐ ⏐ ⏐ ⏐ 4 + 2 √ EX ⏐ ⏐ ⏐ ( µ∗ 0(X) − ˆµ(−k,−k′) 0 (X) )⏐ ⏐ ⏐ 4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ 1 −A 1 −π∗(X) − 1 ] ⏐ ⏐ ⏐...

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(S37) It follows that ( ˆQ(k) )2 −θ2 ETH = ( ˆQ(k) +θETH)( ˆQ(k) −θETH) (i) =Op(1)op(1) = op(1), (S38) where (i) holds from ( S36) and ( S37)

= O(1). (S37) It follows that ( ˆQ(k) )2 −θ2 ETH = ( ˆQ(k) +θETH)( ˆQ(k) −θETH) (i) =Op(1)op(1) = op(1), (S38) where (i) holds from ( S36) and ( S37). According to ( S29), ( S30), and ( S38), ˆB(k) =B +op(1). (S39) Now, we establish the consistency of ˆC (k). By Lemma 9, ˆβ(−k)⊤  E(W ) − |Gk|−1 ∑ i∈Gk Wi   1 ˜n ∑ i∈Ik 2ˆǫi (k) [ ( β∗⊤ Di )2 −θETH ] =o...

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Finally, ∥ǫ∥ψ2 = ∥ϕ∗(Z) − W ⊤β∗∥ψ2 ≤ ∥ϕ∗(Z)∥ψ2 + ∥ W ⊤β∗∥ψ2 (i) ≤O(1) +σx ∥ β∗∥ 2 =O(1), where (i) holds from ∥ϕ∗(Z)∥ψ2 =O(1) and Assumption 6

Hence, ∥ β∗∥ 2 =O(1). Finally, ∥ǫ∥ψ2 = ∥ϕ∗(Z) − W ⊤β∗∥ψ2 ≤ ∥ϕ∗(Z)∥ψ2 + ∥ W ⊤β∗∥ψ2 (i) ≤O(1) +σx ∥ β∗∥ 2 =O(1), where (i) holds from ∥ϕ∗(Z)∥ψ2 =O(1) and Assumption 6. 48 Proof of Lemma 7. By Jensen’s inequality, sup ∥ α∥ 2≤1 |E[W ⊤α]| ≤ sup∥ α∥ 2=1 E|W ⊤α| = O(1), where the last step holds by Assumption 6 and Lemma 1. Since any bounded variable is sub-Gaus...

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[6] [6]

Additionally, 1 ˜n ∑ i∈Ik [ ˆϕ(−k)(Zi) −ϕ∗(Zi) ] 4 =op(1) (S45) 63 holds from Lemma 7 and Markov’s inequality. Therefore, 1 ˜n ∑ i∈Ik [ 2 ( ˆǫ(k) i −ǫi ) β∗⊤ Di ] 2 = 4 ˜n ∑ i∈Ik [ ˆϕ(−k)(Zi) −ϕ∗(Zi) +τ − ˆτpara − ˆβ(−k)⊤ ˆD(k) i + β∗⊤ Di ] 2( β∗⊤ Di )2 =O   4 ˜n ∑ i∈Ik {[ ˆϕ(−k)(Zi) −ϕ∗(Zi) ] 2 + (τ − ˆτpara)2 + ( β∗⊤ Di − ˆβ(−k)⊤ ˆD(k) i )2} ( β∗⊤ Di ...

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Since E(a −b)2 ≤ 2E(a2 +b2), we have EX(∆2

log(p) n ) . Since E(a −b)2 ≤ 2E(a2 +b2), we have EX(∆2

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Hence, EX          1 ∑ k′̸=k |Ik′| ∑ k′̸=k ∑ i∈Ik′ /BD Ai=a∆2iWi       2 ∞    =Op ( sγ log(d) log(p) nN ) =op ( log(p) n ) , since sγ log(d) = o(N)

≤ 2EX ⏐ ⏐ ⏐ ⏐ ⏐[Y (1) −µ∗ 1(X)] [ A ˆπ(−k,−k′)(X) − A π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 + 2EX ⏐ ⏐ ⏐ ⏐ ⏐[Y (0) −µ∗ 0(X)] [ 1 −A 1 − ˆπ(−k,−k′)(X) − 1 −A 1 −π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 2 ≤ 2 √ EX|A [Y (1) −µ∗ 1(X)] |4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ 1 ˆπ(−k,−k′)(X) − 1 π∗(X) ] ⏐ ⏐ ⏐ ⏐ ⏐ 4 + 2 √ EX|(1 −A) [Y (0) −µ∗ 0(X)] |4    √ EX ⏐ ⏐ ⏐ ⏐ ⏐ [ 1 1 − ˆπ(−k,−k′)(X) − 1 1 −π∗(X) ] ⏐ ⏐ ⏐ ⏐ ...

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