Asymptotic Bayes Optimality Under Sparsity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure
Abstract
In this article, we investigate the asymptotic Bayes optimality under sparsity (ABOS) of the Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure (Gavrilov et al., 2009) in the sparse Gaussian sequence model. While the asymptotic optimality properties of the Benjamini-Hochberg procedure have been extensively studied, corresponding results for the GBS procedure remain unavailable despite its favorable finite-sample performance in several settings. Within the spike-and-slab Bayesian formulation and the asymptotic decision-theoretic framework of Bogdan et al. (2011), we establish that the GBS procedure is ABOS over a broad class of sparse asymptotic regimes. Existing analyses of the Benjamini-Hochberg procedure rely on approximating the random rejection threshold by a suitable deterministic surrogate. In contrast, our approach proceeds through a direct analysis of the Bayes risk by separately controlling the false discovery and false nondiscovery components. The analysis combines two key ingredients: a new finite-sample inequality for shifted order statistics associated with the GBS critical constants and a signal-crossing argument for ordered alternative p-values that exploits the sequential step-down structure of the procedure. For diverging expected signal counts, these tools yield ABOS without threshold localization. For bounded positive expected signal counts, a direct argument based on the first GBS critical constant establishes ABOS as well. In particular, our results cover polynomial sparsity pmasymp m−β for every fixed 0<β≤1. To the best of our knowledge, this is the first asymptotic decision-theoretic analysis of the GBS procedure and the first proof of its asymptotic Bayes optimality under sparsity.
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BibTeX
@misc{ghosh2026asymptotic,
title = {{Asymptotic Bayes Optimality Under Sparsity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure}},
author = {Prasenjit Ghosh and Arijit Chakrabarti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.03719},
eprint = {2607.03719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03719},
}
RIS
TY - UNPB
TI - Asymptotic Bayes Optimality Under Sparsity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure
AU - Prasenjit Ghosh
AU - Arijit Chakrabarti
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2607.03719
UR - https://arxiv.org/abs/2607.03719
AB - In this article, we investigate the asymptotic Bayes optimality under sparsity (ABOS) of the Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure (Gavrilov et al., 2009) in the sparse Gaussian sequence model. While the asymptotic optimality properties of the Benjamini-Hochberg procedure have been extensively studied, corresponding results for the GBS procedure remain unavailable despite its favorable finite-sample performance in several settings. Within the spike-and-slab Bayesian formulation and the asymptotic decision-theoretic framework of Bogdan et al. (2011), we establish that the GBS procedure is ABOS over a broad class of sparse asymptotic regimes. Existing analyses of the Benjamini-Hochberg procedure rely on approximating the random rejection threshold by a suitable deterministic surrogate. In contrast, our approach proceeds through a direct analysis of the Bayes risk by separately controlling the false discovery and false nondiscovery components. The analysis combines two key ingredients: a new finite-sample inequality for shifted order statistics associated with the GBS critical constants and a signal-crossing argument for ordered alternative p-values that exploits the sequential step-down structure of the procedure. For diverging expected signal counts, these tools yield ABOS without threshold localization. For bounded positive expected signal counts, a direct argument based on the first GBS critical constant establishes ABOS as well. In particular, our results cover polynomial sparsity p_masymp m^−β for every fixed 0<β≤1. To the best of our knowledge, this is the first asymptotic decision-theoretic analysis of the GBS procedure and the first proof of its asymptotic Bayes optimality under sparsity.
ER -