Probability Theory/Statistics
Chung–Smirnov property for smoothed distribution function estimator under random censorship
[Propriété de Chung–Smirnov de l'estimateur lissé de la fonction de répartition en présence de censure]

Présenté par : Paul Deheuvels

Ould-Saïd, Elias1 ; Yazourh-Benrabah, Ouafae1
1 Université du Littoral Côte d'Opale, LMPA, centre de la Mi-Voix, BP 699, 62228 Calais, France
Comptes Rendus. Mathématique, Tome 337 (2003) no. 3, pp. 207-212.

On considère une suite de variables aléatoires iid (Xn)n⩾1 de même fonction de répartition (f.d.r.) F et une autre suite de variables aléatoires (Cn)n⩾1 de f.d.r. G indépendantes de (Xn)n⩾1. On considère un estimateur lissé par convolution de F. Nous montrons que cet estimateur vérifie la propriété de Chung–Smirnov. Dans cette Note, nous étendons les résultats de Winter (1979) et Degenhardt (1993) au cas censuré et celui de Csörgö et Horvath (1983) à l'estimateur lissé avec la même constante CF,G.

Let (Xn)n⩾1 be a sequence of independent and identically distributed (iid) random variables (rv) with common distribution function (df) F and another iid sequence (Cn)n⩾1 with df G independent of (Xn)n⩾1. Here we consider the Smoothed Kaplan–Meier Estimator of F defined as integral of nonparametric density estimators. It is shown that if F satisfies some smoothness conditions, has the Chung–Smirnov property, that is, with probability one,

where CF,G is a constant depending only on F and G (∥·∥T and T are defined below). In this Note, we extend the result of Winter (1979) and Degenhardt (1993) to the censorship model and those of Csörgö and Horvath (1983) to the smoothed estimator with the same constant CF,G.

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Accepté le :
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DOI : 10.1016/S1631-073X(03)00311-X
Ould-Saïd, Elias 1 ; Yazourh-Benrabah, Ouafae 1

1 Université du Littoral Côte d'Opale, LMPA, centre de la Mi-Voix, BP 699, 62228 Calais, France 
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Ould-Saïd, Elias; Yazourh-Benrabah, Ouafae. Chung–Smirnov property for smoothed distribution function estimator under random censorship. Comptes Rendus. Mathématique, Tome 337 (2003) no. 3, pp. 207-212. doi : 10.1016/S1631-073X(03)00311-X. http://www.numdam.org/articles/10.1016/S1631-073X(03)00311-X/

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