Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### An adaptive test for zero mean

Auteur(s):

Code(s) de Classification MSC:

• 62G10 Hypothesis testing
• 62G08 Nonparametric regression

Résumé: Assume one observes a random vector $y$ of $\R^n$, and write $y=f+\eps$ where $f$ is the expectation of $y$ and $\eps$ is an unobservable centered random vector. The aim of this paper is to build a new test for the null hypothesis that $f= 0$ under as few assumptions as possible on $f$ and $\eps$. The proposed test is nonparametric (no prior assumption on $f$ is needed) and non asymptotic. It has the prescribed level $\alpha$ under the only assumption that the components of $\eps$ are mutually independent, almost surely different from zero, and with symmetrical distribution. Its power is described in a general setting and also in the regression setting, where $f_i=F(x_i)$ for some unknown regression function $F$ and some fixed design points $x_i\in[0,1]$. The test is shown to be adaptive over a H\"olderian smoothness class in the regression setting, under mild assumptions on $\eps$. In particular, we prove adaptive properties of the test when the $\eps_i$'s are not assumed Gaussian nor identically distributed.

Mots Clés: adaptive test ; minimax hypothesis testing ; nonparametric alternatives ; symmetrization ; heteroscedasticity

Date: 2004-05-24

Prépublication numéro: PMA-915