On consistency of kernel density estimators for randomly censored data:: Rates holding uniformly over adaptive intervals

In the usual right-censored data situation, let f(n), n is an element of N, denote the convolution of the Kaplan-Meier product limit estimator with the kernels a(n)(-1) K(./a(n)), where K is a smooth probability density with bounded support and a(n) --> 0. That is, f(n) is the usual kernel density estimator based on Kaplan-Meier. Let (f) over bar (n) denote the convolution of the distribution of the uncensored data, which is assumed to have a bounded density, with the same kernels. For each n, let J(n) denote the half line with right end point Z(n(1-epsilonn)),(n) - a(n), where epsilon (n) --> 0 and, for each m, Z(m,n) is the mth order statistic of the censored data. It is shown that, under some mild conditions on a(n) and epsilon (n) sup(Jn) / f(n) (t) - (f) over bar (n)(t)/ converges a.s. to zero as n --> infinity at least as fast as root /log(a(n) boolean AND epsilon (n))//(na(n)epsilon (n)). For epsilon (n) = constant, this rate compares, up to constants, with the exact rate for fixed intervals. (C) 2001 Editions scientifiques et medicales Elsevier SAS.