On Consistency of Kernel Density Estimators for Randomly Censored Data: Rates Holding Uniformly over Adaptive Intervals

In the usual right-censored data situation, let fn, n∈N, denote the convolution of the Kaplan-Meier product limit estimator with the kernels a−1 n K(·/an), where K is a smooth probability density with bounded support and an→0. That is, fn is the usual kernel density estimator based on Kaplan-Meier. Let f̄n 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 Jn denote the half line with right end point Zn,n(1−εn)−an, where εn→0 and, for each m, Zn,m is the m-th order statistic of the censored data. It is shown that, under some mild conditions on an and εn, supJn |fn(t)−f̄n(t)| converges a.s. to zero as n→∞ at least as fast as √ | log(an∧εn)|/(nanεn). For εn=constant, this rate compares, up to constants, with the exact rate for fixed intervals.