Validity of the Hochberg procedure under jointly Gaussian dependence

Establish whether the Hochberg multiple-testing procedure, which is based on the Simes test, controls the family-wise error rate when applied to collections of positively dependent test statistics that are jointly Gaussian distributed; either prove validity under precise dependence conditions or provide a counterexample demonstrating failure.

Background

In discussing multiple-hypothesis testing corrections for their statistical analysis, the authors note that the Holm–Bonferroni procedure was used, and they consider the Hochberg procedure as a more powerful alternative. The Hochberg procedure relies on the Simes test, which is known to be conservative under certain forms of positive dependence.

However, despite expectations of validity in the jointly Gaussian case, the authors explicitly acknowledge the absence of a proof (or disproof) that the Hochberg procedure controls error rates under jointly Gaussian dependence. This raises a concrete methodological open question of broad relevance to statistical inference in similar high-dimensional, correlated settings.

References

An alternative is the Hochberg procedure, which is more powerful than the Holm–Bonferroni procedure. It is based on the Simes test, which is conservative for tests that are positively dependent in a certain sense. It may be expected to be valid in the case of a jointly Gaussian distribution, but to our knowledge, at this point in time there is no proof of this (or proof to the contrary).

— Quantifying T cell morphodynamics and migration in 3D collagen matrices  (2401.03595 - Low, 2024) in Footnote following Holm–Bonferroni discussion, Section "Dimensionality reduction of the shape of motile cells"

QUDE, the closest prior system, integrates multiple-hypothesis control into interactive data exploration, but it assumes the hypotheses are independent and names dependence as an open limitation (Section~\ref{sec:related}).

— Tacet: A Language and Type System for Automatic Statistical Validity Accounting  (2608.27451 - Abuah, 27 Aug 2026) in Section 1, Introduction; Section 8, Related Work, subsection “Interactive data exploration”