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arXiv:2604.20059 · detector doi_compliance · incontrovertible · 2026-05-20 02:21:58.066213+00:00

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O. V. Lepskii. “Asymptotically Minimax Adaptive Estimation. I: Upper Bounds. Optimally Adaptive Estimates”. In:Theory of Probability & Its Applications36.4 (1992), pp. 682–697.doi:10 . 1137 / 1136085. eprint:https://doi.org/10.1137/1136085.url:https://doi.org/10.1137/1136085. 16 Appendix Auxiliary theoretical analysis Proof.Plug-in variance estimation.Recall the efficient influence function for the ATE: D(O) = X a∈{0,1} I(A=a) g(a|W) (Y− ¯Q(a, W)) + ¯Q(1, W)− ¯Q(0, W)−Ψ. We decompose its variance: Var(D(O)) = X a∈{0,1} Var I(A=a) g(a|W) (Y− ¯Q(a, W)) + Var ¯Q(1, W)− ¯Q(0, W)−Ψ , since the cross-covariance terms vanish. Indeed, E I(A=a)(Y− ¯Q(a, W))|W =g(a|W)E Y− ¯Q(a, W)|A=a, W = 0. For continuous outcomes, E " I(A=a) g(a|W) 2 (Y− ¯Q(a, W))2 # =E I(A=a) g(a|W) 2 (Y− ¯Q(a, W))2 =E W Var(Y|A=a, W) g(a|W) . LetR i =Y i − ¯Q(Ai, Wi). Then E[R2 i |A=a, W] = Var(Y|A=a, W). Estimating this conditional variance by regressingR 2 i onWwithin each treatment arm yields ˆσ 2(a, W). Substituting these estimates into the variance decomposition gives dVarplug-in( ˆψ) = 1 n nX i=1 ˆσ2(1, Wi) ˆg(1|Wi) + ˆσ2(0, Wi) ˆg(0|Wi) + 1 n nX i=1 h ˆQ(1, Wi)− ˆQ(0, Wi)− ˆψ i2 , which establishes the stated plug-in variance representation. Detailed and additional simulation results This section provides detailed simulation results that support the main findings, together with additional simulation comparisons. Figures A.1 to A.17 report the full set of detailed metrics for the adaptive trun- cation proced

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