A generalization of the product-limit estimator with an application to censored regression

Abstract

The product-limit estimator (PLE) and weighted empirical processes are two important ingredients of almost any censored regression analysis. A link between them is provided by the generalized PLEs introduced in this paper. These generalized PLEs are the product-limit integrals of the empirical cumulative hazard function estimators in which weighted empirical processes are used to replace the standard empirical processes. The weak convergence and some large sample approximations of the generalized PLEs are established. As an application these generalized PLEs are used to define some minimum distance estimators which are shown to be asymptotically normal. These estimators are qualitatively robust. In some submodels an optimal choice of the weight matrix is the covariate matrix and some of these estimators are quite efficient at a few common survival distributions. To implement these estimators some computational aspects are discussed and an algorithm is given. From a real data example and some preliminary simulation results, these estimators seem to be very competitive to and more robust than some more traditional estimators such as the Buckley-James estimator.