Chung–Smirnov property for smoothed distribution function estimator under random censorship

Abstract

Let ( X n ) n⩾1 be a sequence of independent and identically distributed (iid) random variables (rv) with common distribution function (df) F and another iid sequence ( C n ) n⩾1 with df G independent of ( X n ) n⩾1 . Here we consider the Smoothed Kaplan–Meier Estimator F ̇ n of F defined as integral of nonparametric density estimators. It is shown that if F satisfies some smoothness conditions, F ̇ n has the Chung–Smirnov property, that is, with probability one, lim sup n→∞ 2n loglogn 1/2∥ F n ̇ −F∥ T=C F,G, where C F, 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 C F, G . To cite this article: E. Ould-Saïd, O. Yazourh-Benrabah, C. R. Acad. Sci. Paris, Ser. I 337 (2003).