Product-limit estimators in the analysis of competing risks

In the case of independent competing risks of death, product-limit (PL) estimators, as explained in the introduction, are derived for the net, crude and partial crude probabilities of death by using the method of maximum likelihood. Such use is made considering all of the causes of death simultaneously, without regarding deaths due to “other causes of death” as withdrawals. The estimators of the probabilities based upon different non-parametric models for competing risks and different actuarial methods of estimation are the PL estimators when the subintervals of the period of observation are so shortened that each interval contains room for either one death or for one or several withdrawals. The estimators, derived from the non-parametric models, converge towards the estimators which result from the model for independent risks. This convergence is in accordance with the convergence property of the corresponding probabilities (Hakulinen, 1977).