Sharp Asymptotic Minimaxity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure in Sparse Gaussian Sequence Models

Abstract

We investigate the sharp asymptotic minimaxity of the classical Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure in sparse Gaussian sequence models. Abraham et al. (2024) recently established sharp asymptotic minimax benchmarks for sparse multiple testing under both the classical beta-min framework and more general heterogeneous signal-strength classes. Although several multiple testing procedures attain these benchmarks, corresponding results for the GBS procedure have remained unavailable. We prove that the GBS procedure is sharply asymptotically minimax under both normalized Hamming loss and the combined $\mathrm{FDP}+\mathrm{FNP}$ loss over the classical beta-min parameter classes. We further establish a sharp asymptotic upper bound for the normalized Hamming risk over the heterogeneous signal-strength classes and show that the GBS procedure attains the corresponding sharp asymptotic minimax benchmark under the combined $\mathrm{FDP}+\mathrm{FNP}$ loss. Our analysis is based on a shifted order-statistic inequality for the GBS critical values together with deterministic and empirical signal-crossing arguments for ordered alternative $p$-values. Unlike existing analyses of Benjamini-Hochberg procedures and empirical Bayes $\ell$-value methods, the proposed approach is intrinsic to the geometry of the GBS step-down procedure and avoids localization of a single implicit rejection threshold. Consequently, our results extend sharp asymptotic minimaxity theory to a genuinely step-down multiple testing procedure whose rejection rule depends on the entire ordered sequence of $p$-values.