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 F-n 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 phi(s)dF(n)(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 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. (C) 2000 Elsevier Science B.V. All rights reserved.Statistics & ProbabilitySCI(E)8ARTICLE3235-2444