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.