A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

In this paper we consider the TJW product-limit estimator F̂_n(x) of an unknown distribution function F when the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimator F̂_n(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation of F̂_n(x) - F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator to F. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given.