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We study nonparametric estimation of the sub-distribution functions forcurrent status data with competing risks. Our main interest is in thenonparametric maximum likelihood estimator (MLE), and for comparison we alsoconsider a simpler ``naive estimator.'' Both types of estimators were studiedby Jewell, van der Laan and Henneman [Biometrika (2003) 90 183--197], butlittle was known about their large sample properties. We have started to fillthis gap, by proving that the estimators are consistent and converge globallyand locally at rate $n^{1/3}$. We also show that this local rate of convergenceis optimal in a minimax sense. The proof of the local rate of convergence ofthe MLE uses new methods, and relies on a rate result for the sum of the MLEsof the sub-distribution functions which holds uniformly on a fixed neighborhoodof a point. Our results are used in Groeneboom, Maathuis and Wellner [Ann.Statist. (2008) 36 1064--1089] to obtain the local limiting distributions ofthe estimators.

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