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 (F) over cap(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 (F) over cap(n)(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation of (F) over cap (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 t o F. A sharp rate of convergence theorem concerning the smoothed TJW produc t-limit estimator is obtained. Asymptotic properties of kernel estimators o f density function based on TJW product-limit estimator is given. (C) 1999 Academic Press. AMS 1991 subject classifications: 62G05, 60F17, 62E20, 62G2 0.