Statistique/Probabilités
Processus de Bickel–Rosenblatt pondéré et tests d'ajustement
[Weighted Bickel–Rosenblatt process and goodness of fit tests]
Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 311-316.

The goal of this work is to establish the limit distribution of the process

 $\left\{{I}_{n}\left(W\right):=\underset{A}{\int }{\left({f}_{n}\left(x\right)-E{f}_{n}\left(x\right)\right)}^{2}W\left(x\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}x,\phantom{\rule{3.30002pt}{0ex}}W\in 𝒲\right\},$
where $𝒲$ is a class of weight functions W, fn is the kernel density estimator of the density f and A is a Borelian subset of $ℝ$. We apply this result to derive new statistics to test goodness-of-fit of the density function f. Under some local alternatives, these new tests are more powerful than the usual Bickel–Rosenblatt one.

Le but de cette étude est de trouver la loi limite du processus

 $\left\{{I}_{n}\left(W\right):=\underset{A}{\int }{\left({f}_{n}\left(x\right)-E{f}_{n}\left(x\right)\right)}^{2}W\left(x\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}x,\phantom{\rule{3.30002pt}{0ex}}W\in 𝒲\right\},$
$𝒲$ est une classe de fonctions de poids W, fn est l'estimateur à noyau de la densité f et A est un sous-ensemble borélien de $ℝ$. On utilise ce résultat pour construire de nouveaux tests d'ajustement de la densité f, plus performants, sous certaines alternatives locales, que le test classique de Bickel–Rosenblatt.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.11.031
Chebana, Fateh 1

1 LSTA, boîte Courrier 158, 8A, Université Paris-6, 175, rue du Chevaleret, 75013 Paris, France
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title = {Processus de {Bickel{\textendash}Rosenblatt} pond\'er\'e et tests d'ajustement},
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Chebana, Fateh. Processus de Bickel–Rosenblatt pondéré et tests d'ajustement. Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 311-316. doi : 10.1016/j.crma.2003.11.031. http://www.numdam.org/articles/10.1016/j.crma.2003.11.031/

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