The Annals of Statistics

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

Piet Groeneboom, Marloes H. Maathuis, and Jon A. Wellner

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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 n1/3. 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.

Article information

Ann. Statist. Volume 36, Number 3 (2008), 1031-1063.

First available in Project Euclid: 26 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62N01: Censored data models 62G20: Asymptotic properties
Secondary: 62G05: Estimation

Survival analysis current status data competing risks maximum likelihood consistency rate of convergence


Groeneboom, Piet; Maathuis, Marloes H.; Wellner, Jon A. Current status data with competing risks: Consistency and rates of convergence of the MLE. Ann. Statist. 36 (2008), no. 3, 1031--1063. doi:10.1214/009053607000000974.

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