## Bernoulli

- Bernoulli
- Volume 20, Number 1 (2014), 304-333.

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

Gilles Blanchard, Sylvain Delattre, and Etienne Roquain

#### 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.

#### Article information

**Source**

Bernoulli, Volume 20, Number 1 (2014), 304-333.

**Dates**

First available in Project Euclid: 22 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1390407291

**Digital Object Identifier**

doi:10.3150/12-BEJ488

**Mathematical Reviews number (MathSciNet)**

MR3160584

**Zentralblatt MATH identifier**

06282553

**Keywords**

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

#### Citation

Blanchard, Gilles; Delattre, Sylvain; Roquain, Etienne. Testing over a continuum of null hypotheses with False Discovery Rate control. Bernoulli 20 (2014), no. 1, 304--333. doi:10.3150/12-BEJ488. https://projecteuclid.org/euclid.bj/1390407291

#### Supplemental materials

- Supplementary material: Supplement to: “Testing over a continuum of null hypotheses with False Discovery Rate control”. This supplement provides some technical results and introduces the so-called general PRDS condition, which is a stronger assumption than the finite dimensional PRDS condition. This condition is useful to prove FDR control for procedures which are not necessarily of the step-up type.Digital Object Identifier: doi:10.3150/12-BEJ488SUPP