Exact Simes inequalities for two-sided Gaussian tests

Abstract

The Simes test combines marginal p-values to test a global null hypothesis. Whether it controls the type I error for two-sided Gaussian tests with arbitrary covariance remains open. We prove that, for every fixed number $m$ of tests, the rejection probability does not exceed the nominal level when that level is sufficiently small. The same range of levels works for every covariance matrix. Equality holds if and only if the p-values form independent groups with identical values within each group. For $m=3$, a computer-assisted proof establishes validity at $0.05$, and an analytic argument extends it to an explicit interval around $0.05$. As the number $m$ of tests increases, we prove validity in two settings: each test statistic is correlated with at most a fixed number of others; or the statistics form independent blocks whose sizes grow sufficiently slowly, at a specified rate of order $\log\log m$. Both settings allow arbitrary correlations within these restrictions. For any fixed $0<a<1$, validity holds for every $m$ above a threshold $M$ and every level $0<α\le a$; the same $M$ works for all allowed covariances. Further results quantify the difference between the nominal level and the rejection probability, and permit more correlated neighbors when their correlations are bounded away from $\pm1$. We do not obtain numerical values for the general small-level cutoffs or the required number of tests.