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

In this article, we consider a useful product-limit estimator of distribution function proposed by Huang & Qin(2011) when the observations are subject to length-biased and right-censored data. The estimator retains the simple closed-form expression of the truncation product-limit estimator with some good properties. An almost sure representation for the estimator is obtained which can be used to derive many properties of functional statistics based on this product-limit estimator. The rate for the remainder in the representation is of order $O(n^{-3/4}(\log(n)^{3/4}) $ a.s.