Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Béatrice Laurent (1) , Jean-Michel Loubes (1) , Clément Marteau (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

The aim of this paper is to establish non-asymptotic minimax rates of testing for goodness-of-fit hypotheses in a heteroscedastic setting. More precisely, we deal with sequences $(Y_j)_{j\in J}$ of independent Gaussian random variables, having mean $(\theta_j)_{j\in J}$ and variance $(\sigma_j)_{j\in J}$. The set $J$ will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to $l_2$ and $l_{\infty}$ norms, without assumption on $(\sigma_j)_{j\in J}$ and on several functions spaces. Our point of view is completely non-asymptotic.

Mots clés: Goodness-of-fit tests; heterogeneous variances; inverse problems.; inverse problems

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Béatrice Laurent, Jean-Michel Loubes, Clément Marteau. Non asymptotic minimax rates of testing in signal detection with heterogeneous variances. Electronic Journal of Statistics , 2012, 6, pp.91-122. ⟨hal-00440825v2⟩