ar X iv : m at h / 06 09 02 1 v 1 [ m at h . ST ] 1 S ep 2 00 6 CURRENT STATUS DATA WITH COMPETING RISKS : LIMITING DISTRIBUTION OF THE MLE

Delft University of Technology and Vrije Universiteit Amsterdam, University of Washington and University of Washington We study nonparametric estimation 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 [10]. Groeneboom, Maathuis and Wellner [7] established that both estimators converge globally and locally at rate n. In this paper we use these results to derive the local limiting distributions of the estimators. The limiting distribution of the naive estimator is given by the slopes of the convex minorants of correlated Brownian motion processes with parabolic drifts. The limiting distribution of the MLE involves a new self-induced process. We prove that this process exists and is almost surely unique. Finally, we present a simulation study showing that the MLE is superior to the naive estimator in terms of mean squared error, both for small sample sizes and asymptotically. Supported in part by NSF grant DMS-0203320 Supported in part by NSF grants DMS-0203320 and DMS-0503822 and by NI-AID grant 2R01 AI291968-04 AMS 2000 subject classifications: Primary 62N01, 62G20; secondary 62G05