The strong representation for the nonparametric estimator of length-biased and right-censored data

In this paper, we consider the modified product-limit estimator of an unknown distribution function proposed by Huang and Qin (2011), where the observations are subject to length-biased and right-censored data. A strong representation for the modified product-limit estimator is established with a remainder O(n−3/4(logn)3/4)O(n−3/4(logn)3/4) a.s. Such results are very useful when we consider statistics that are the function of the estimator of nonparametric distribution function. Also, a uniform consistency rate of the estimator is given.