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

Piet Groeneboom·Marloes H. Maathuis·Jon A. Wellner
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Abstract

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 a simpler ``naive estimator.'' Both types of estimators were studied by Jewell, van der Laan and Henneman [Biometrika (2003) 90 183--197], but little was known about their large sample properties. We have started to fill this gap, by proving that the estimators are consistent 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 [Ann. Statist. (2008) 36 1064--1089] to obtain the local limiting distributions of the estimators.

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Piet Groeneboom, Marloes H. Maathuis, Jon A. Wellner. 2008-06-17. Current status data with competing risks: Consistency and rates of Convergence of the MLE. https://doi.org/10.1214/009053607000000974

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