Simes error control for two-sided Gaussian tests
Let $n \ge 1$ and let $\Sigma$ be a real symmetric positive semidefinite $n \times n$ matrix with unit diagonal. Let $X = (X_1, \dots, X_n) \sim N_n(0, \Sigma)$ and let $P_i = 2\Phi(-|X_i|)$ be the two-sided Gaussian $p$-values, where $\Phi$ is the standard normal distribution function, so that every null hypothesis is true. With $P_{(1)} \le \dots \le P_{(n)}$ the order statistics, let $$S_n = \min_{1 \le k \le n} \frac{n P_{(k)}}{k}$$ be the Simes statistic. Simes showed that $\Pr\{S_n \le \alpha\} = \alpha$ when the $P_i$ are independent. Sarkar proved the inequality below for $p$-values that are multivariate totally positive of order two (MTP$_2$), and by Karlin and Rinott $(|X_1|, \dots, |X_n|)$ is MTP$_2$ when $\Sigma$ is positive definite and the off-diagonal entries of $-D\Sigma^{-1}D$ are nonnegative for some diagonal matrix $D$ with diagonal entries $\pm 1$, which gives the inequality for such $\Sigma$ (Sarkar, 2008). For one-sided Gaussian $p$-values with a negative correlation the inequality can fail (Hochberg and Rom; see Chi, Ramdas and Wang). When every null hypothesis is true, the false discovery rate of the Benjamini–Hochberg procedure at level $\alpha$ equals $\Pr\{S_n \le \alpha\}$; Dobriban showed that for two-sided Gaussian $p$-values with some nonzero means that procedure can fail to control the false discovery rate (2026).
Problem. Is $\Pr\{S_n \le \alpha\} \le \alpha$ for every $n \ge 1$, every such $\Sigma$ and every $\alpha \in (0,1)$?
Source: A well-known conjecture (Chi, Ramdas and Wang (2025), Remark 11), attributed by Sarkar (2023), Section 1, to Finner, Roters and Strassburger (2017).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- ST — Statistics Theory
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, multiple testing, Simes inequality, two-sided tests
References
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author = {Chi, Ziyu and Ramdas, Aaditya and Wang, Ruodu},
title = {Multiple testing under negative dependence},
journal = {Bernoulli},
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year = {2025},
pages = {1230--1255},
doi = {10.3150/24-BEJ1768},
eprint = {2212.09706},
archivePrefix = {arXiv},
}
@article{finner2017,
author = {Finner, Helmut and Roters, Markus and Strassburger, Klaus},
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doi = {10.1007/s00362-015-0725-8},
}
@misc{sarkar2023,
author = {Sarkar, Sanat K.},
title = {On controlling the false discovery rate in multiple testing of the means of correlated normals against two-sided alternatives},
year = {2023},
eprint = {2304.05261},
archivePrefix = {arXiv},
}
@article{simes1986,
author = {Simes, R. J.},
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}
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author = {Sarkar, Sanat K.},
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@incollection{sarkar2008,
author = {Sarkar, Sanat K.},
title = {On the {S}imes inequality and its generalization},
booktitle = {Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen},
series = {Inst. Math. Stat. Collect.},
volume = {1},
publisher = {Institute of Mathematical Statistics},
address = {Beachwood, OH},
year = {2008},
pages = {231--242},
doi = {10.1214/193940307000000167},
eprint = {0805.2322},
archivePrefix = {arXiv},
}
@misc{dobriban2026,
author = {Dobriban, Edgar},
title = {The {B}enjamini--{H}ochberg procedure can fail to control the {FDR} for correlated two-sided {G}aussian tests},
year = {2026},
eprint = {2607.12208},
archivePrefix = {arXiv},
}
@article{hochberg1995,
author = {Hochberg, Yosef and Rom, Dror},
title = {Extensions of multiple testing procedures based on {S}imes' test},
journal = {J. Statist. Plann. Inference},
volume = {48},
number = {2},
year = {1995},
pages = {141--152},
doi = {10.1016/0378-3758(95)00005-T},
}
@misc{aim-two-sided-gaussian-simes,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Simes error control for two-sided Gaussian tests}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/329-two-sided-gaussian-simes.md},
note = {Entry 329 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00311, "Simes error control for two-sided Gaussian tests", clopen.solutions.