Asymptotic Optimality of the Product Limit Estimator

Abstract

The product limit estimator due to Kaplan and Meier (1958) is well-known to be the nonparametric maximum likelihood estimator of a distribution function based on censored data. It is shown here that the product limit estimator is an asymptotically optimal estimator in two senses: in the sense of a Hajek-Beran type representation theorem for regular estimators; and in an asymptotic minimax sense similar to the classical result for the uncensored case due to Dvoretzky, Kiefer, and Wolfowitz (1956). The proofs rely on the methods of Beran (1977) and Millar (1979).