Finite-sample nonparametric mean tests: Leave-one-out duality and asymptotic optimality
Abstract
We study finite-sample valid tests of the one-sided mean hypothesis H0:μ≤ 1 against H1:μ>1 for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null 𝔼[Xi|X−i]≤ 1, and which also gives conditions that allow combining dual certificates for p-values to show that their pointwise minimum is also a valid p-value. The framework proves finite-sample validity of Wang and Zhao's nonparametric likelihood-ratio statistic Tnplr, yields a new p-value Tbin+ extending the Clopper--Pearson binomial test to general nonnegative random variables, and shows that the pointwise minimum min{Tnplr,Tbin+} is itself a valid and more powerful p-value. We establish sharp optimality results for such testing problems in two regimes: both Tnplr and Tbin+ attain a universal detectability boundary for the null H0 without moment or tail assumptions, and Tbin+ attains a nonparametric power lower bound under n−1/2-local alternatives to H0. Efficient algorithms and numerical experiments demonstrate substantial finite-sample power gains over existing valid methods.
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BibTeX
@misc{zhu2026finitesample,
title = {{Finite-sample nonparametric mean tests: Leave-one-out duality and asymptotic optimality}},
author = {Yifan Zhu and John C. Duchi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05360},
eprint = {2609.05360},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05360},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-sample nonparametric mean tests: Leave-one-out duality and asymptotic optimality
AU - Yifan Zhu
AU - John C. Duchi
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05360
UR - https://arxiv.org/abs/2609.05360
AB - We study finite-sample valid tests of the one-sided mean hypothesis H_0:μ≤ 1 against H_1:μ>1 for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null 𝔼[X_i|X_−i]≤ 1, and which also gives conditions that allow combining dual certificates for p-values to show that their pointwise minimum is also a valid p-value. The framework proves finite-sample validity of Wang and Zhao's nonparametric likelihood-ratio statistic T_nplr, yields a new p-value T_bin+ extending the Clopper--Pearson binomial test to general nonnegative random variables, and shows that the pointwise minimum min{T_nplr,T_bin+} is itself a valid and more powerful p-value. We establish sharp optimality results for such testing problems in two regimes: both T_nplr and T_bin+ attain a universal detectability boundary for the null H_0 without moment or tail assumptions, and T_bin+ attains a nonparametric power lower bound under n^−1/2-local alternatives to H_0. Efficient algorithms and numerical experiments demonstrate substantial finite-sample power gains over existing valid methods.
ER -