Current status data with competing risks: consistency and rates of convergence of the MLE

We study nonparametric estimation of the sub-distribution func-tions for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider a simpler ‘naive estimator’. Both types of estimators were studied by Jewell, Van der Laan and Henne-man [7], but little was known about their large sample properties. We have started to fill this gap, by proving that the estimators are consis-tent and converge globally and locally at rate n 1/3. 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. Our results are used in Groeneboom, Maathuis and Wellner [3] to obtain the local limiting distributions of the estimators