the extended glivenko-cantelli property for kernel-smoothed estimator of the cumulative distribution function in the length-biased sampling

When the probability of selecting an individual from a population is proportional to its length, the resulting distribution of observation will exhibit length bias. this distribution is referred to as a length-biased distribution. let {yi;i = 1, . . . , n} be a sample from a length-biased population with cumulative distribution function g(·). in this paper we consider cox’s empirical estimator f c n(·) and the smoothed kernel-type estimator f s n(·) of f(·). under suitable conditions, the extended glivenko-cantelli theorem for f c n(·) and f s n(·) are proved. also, the validity of the extended glivenko-cantelli property for the smoother estimator f s n(·) is investigated using a simulation study.