The Admissibility of the Kaplan-Meier and other Maximum Likelihood Estimators in the Presence of Censoring

Abstract

For the nonparametric estimation of a survival function when censoring is present, the Kaplan-Meier estimator is often used. The admissibility of this estimator and other related maximum likelihood estimators is demonstrated. This is done by reducing the problem to one involving just the multinomial distribution and then using the stepwise Bayes technique to prove admissibility. 1. Introduction. The admissibility of various well-known nonparametric estimators was recently demonstrated in Meeden, Ghosh and Vardeman (1985). The argument proving admissibility involved two essential steps. The first was to note that to prove admissibility for the nonparametric problem it was enough to prove admissibility when the family of possible distributions was just assumed to be some multinomial family of distributions, rather than the large nonparametric family. The second step then used the stepwise Bayes technique to prove admissibility for the multinomial problem. Here we will use the same argument to prove the admissibility of the Kaplan-Meier estimator for the nonparametric estimation of the survival function when censoring is present. In fact, the admissibility of a whole class of maximum likelihood estimators, which includes the Kaplan-Meier estimator, is demonstrated. The only additional complication is that the censoring mechanism must be modeled and then taken into account. This description and modeling of the censoring is carried out in Section 3. In Section 4, the stepwise Bayes technique is used to demonstrate the admissibility of the maximum likelihood estimator for the multinomial problem. In Section 5, the admissibility results for the nonparametric problem are given. If we were just proving the admissibility of the Kaplan-Meier estimator this paper would be considerably shorter. Although the proofs for the Kaplan-Meier estimator and the more general case are essentially the same, the general case requires considerably more notation. For someone who is just interested in the Kaplan-Meier estimator the argument can be simplified using the notation of