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.