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

### Nonparametric homogeneity tests

Auteur(s):

Code(s) de Classification MSC:

• 62G10 Hypothesis testing
• 62H10 Distribution of statistics
• 62H15 Hypothesis testing

Résumé: We test whether two independent samples of i.i.d. random variables $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_m$ having common probability density $f$ and, respectively, $g$ are issued from the same population. The null hypothesis $f=g$ is opposed to a large nonparametric class of smooth alternatives $f$ and $g$. We consider several problems, according to the distance between the populations' densities: pointwise, interval-wise, $L_2$ and $L_\infty$ norms. We propose test procedures that attain parametric rates in some cases. In other problems, the procedures adapt automatically to the smoothnesses of the underlying densities. After a numerical study of these tests, we prove their theoretical properties in the classical minimax approach.

Mots Clés: Nonparametric test ; Homogeneity test ; Wavelet estimator ; Minimax rates ; Adaptivity

Date: 2003-12-15

Prépublication numéro: PMA-871