On the strong convergence of the product-limit estimator and its integrals under censoring and random truncation

It is shown that the product limit estimator Fn of a continuous distribution function F based on the right censored and left truncated data is uniformly strong consistent over the entire observation interval of F allowed under censoring and truncation. Moreover, it is shown that the integral [integral operator] [phi](s) dFn(s) converges almost surely as n-->[infinity] for any nonnegative measurable function [phi] satisfying some mild conditions. The limits of these integrals, however, need not be [integral operator] [phi](s) dF(s). The results are important for studying convergence of sample moments and regression problems when both censoring and truncation are present. A condition of identifiability, often overlooked in the literature is discussed.