Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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*