Current Status Data with Competing Risks: Consistency and Rates of Convergence of the Mle

Delft University of Technology and Vrije Universiteit Amsterdam, University of Washington and University of Washington We study nonparametric estimation of the sub-distribution functions for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider the ‘naive estimator’ of Jewell, Van der Laan and Henneman [9]. We prove that both estimators are consistent and converge globally and locally at rate n. We also show that this local rate of convergence is optimal in a minimax sense. The proof of the local rate of convergence of the MLE uses new methods, and relies on a rate result for the sum of the MLEs of the sub-distribution functions which holds uniformly on a fixed neighborhood of a point. The results of this paper are used in Groeneboom, Maathuis and Wellner [5] to derive the local limiting distributions of the estimators.