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

Gilles Blanchard (1) , Sylvain Delattre (2) , Etienne Roquain (2)

1 Institut für Mathematik [Potsdam]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Keywords: continuous testing; stochastic process; multiple testing; false discovery rate; step-up; positive correlation

Domains: Methodology [stat.ME]

Gilles Blanchard, Sylvain Delattre, Etienne Roquain. Testing over a continuum of null hypotheses with False Discovery Rate control. Bernoulli, 2014, pp.304-333. ⟨10.3150/12-BEJ488⟩. ⟨hal-00632783v3⟩