A key identity for the product-limit estimator due to O. O. Aalen and S. Johansen [Scand. J. Stat., Theory Appl. 5, 141-150 (1978; Zbl 0383.62058)] and R. D. Gill [Censoring and stochastic integrals. Math. Centre Tracts 124 (1980; Zbl 0456.62003)] is shown to be a consequence of the exponential formula of C. Doléans-Dade [Z. Wahrscheinlichkeitstheor. Verw. Geb. 16, 181-194 (1970; Zbl 0194.491)].
The basic counting processes in the censored data problem are shown to converge jointly to Poisson processes under ”heavy-censoring”: \(G_ n\to_ d\delta_ 0\), but \(n(1-G_ n)\to \alpha\) 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 Y. Y. Chen, M. Hollander and N. A. Langberg [J. Am. Stat. Assoc. 77, 141-144 (1982; Zbl 0504.62033)].