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

Abstract

False discovery rate control does not by itself guarantee optimal recovery of sparse signals: a procedure may make few false discoveries while missing an unnecessarily large fraction of the alternatives. We investigate whether the classical Gavrilov--Benjamini--Sarkar (GBS) step-down procedure also achieves the sharp frequentist minimax limits for sparse recovery in independent Gaussian sequence models. Under a beta-min separation at the critical sparse detection boundary, GBS attains the exact asymptotic risk constant $\bar{\Phi }(b)$ for normalized Hamming loss and combined $\mathrm{FDP}+\mathrm{FNP}$ loss. The Hamming conclusion remains valid when the number of signals is only bounded above by the nominal sparsity $s_n$. For heterogeneous signal magnitudes, the corresponding risk under either loss equals the sharp benchmark ${\Lambda }_n(a)+o(1)$, without assuming that ${\Lambda }_n(a)$ converges; this is additive, rather than necessarily relative, sharpness. We assume $s_n\to \infty$, $n/s_n\to \infty$, and a vanishing nominal level ${\alpha }_n$ satisfying $\log (1/{\alpha }_n)=o(\sqrt{\log (n/s_n)})$. Unlike step-up rules, a step-down procedure must satisfy every preceding ordered $p$-value constraint before reaching a given rank. Our analysis handles this sequential obstruction directly: a finite-sample shifted order-statistic bound makes false positives negligible, while new uniform crossing arguments show that the rejection path reaches the optimal fraction of signals. The results extend the range of testing architectures known to attain the sharp minimax benchmarks, without requiring localization of a single rejection threshold.