A Heavy Censoring Limit Theorem for the Product Limit Estimator

Abstract

A key identity for the product-limit estimator due to Aalen and Johansen (1978) and Gill (1980) is shown to be a consequence of the exponential formula of Doleans-Dade (1970). The basic counting processes in the censored data problem are shown to converge jointly to Poisson processes under "heavy-censoring": $G_n \rightarrow_d \delta_0$, but $n(1 - G_n) \rightarrow \alpha$ where $G_n$ is the censoring distribution. The Poisson limit theorem for counting processes implies Poisson type limit theorems under heavy censoring for the cumulative hazard function estimator and product limit estimator. The latter, in combination with the key identity of Aalen-Johansen and Gill and martingale properties of the limit processes, yields a new approximate variance formula for the product limit estimator which is compared numerically with recent finite sample calculations for the case of proportional hazard censoring due to Chen, Hollander, and Langberg (1982).