Testing over a continuum of null hypotheses with False Discovery Rate control

TL;DR

This paper extends false discovery rate control to a broad setting of infinitely many hypotheses, providing procedures that work under dependence conditions, with applications in non-parametric testing scenarios.

Contribution

It generalizes FDR control methods to uncountably infinite hypothesis sets using $p$-value processes, under dependence assumptions, broadening applicability.

Findings

FDR control achieved under arbitrary dependence of $p$-values.

FDR control established under positive dependence (weak PRDS).

Applications demonstrated in Gaussian, Poisson, and i.i.d. testing scenarios.

Abstract

We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding $p$-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the $p$-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of $p$-values, or under the assumption that the finite dimensional distributions of the $p$-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.