Estimating Dependent Life Lengths, with Applications to the Theory of Competing Risks

Abstract

In the classical theory of competing risks (as well as in many reliability models and incomplete data problems) it is assumed that (A1) the risks (i.e., the random variables of interest) are independent and that (A2) death does not result from simultaneous causes. Employing our probabilistic solution to a related problem in probability modelling, we obtain strongly consistent estimators for the unobservable marginal distributions of interest. These estimators are analogous to those of Kaplan and Meier [J. Amer. Statist. Assoc. (1958) 63] and are appropriate when the assumptions of independence and no simultaneous causes of death [(A1) and (A2), above] fail to hold. We show how our methods can be used to unify and simplify the nonparametric approach toward estimation in the competing risks model. As a consequence we obtain an elementary proof of the strong consistency of the Kaplan-Meier estimator. Our results extend and simplify the work of Peterson [J. Amer. Statist. Assoc. (1977) 72] and Desu and Narula [The Theory and Applications of Reliability, I (I. Shimi and C. P. Tsokos, eds.) (1977)].