The extended Glivenko-Cantelli property for Kernel-Smoothed estimator of the cumulative distribution function in the length-biased sampling

‎Let \{Y_i; i = ۱,\ldots,n \} be a length-biased sample from a population with cumulative distribution function F(\cdot)‎. If the probability of an item selected in the sample is proportional to its length‎, ‎then the distribution of the observed length is known as the length-biased distribution‎.‎We consider the kernel-type estimator F_n^s(\cdot) of F(\cdot)‎. ‎Under suitable conditions‎, ‎the extended Glivenko-Cantelli theorem for F_n^s(\cdot) is proved‎.

کلیدواژه ها:

Extended Glivenko-Cantelli theorem ، Smoothed estimator ، Strong consistency