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Current status data with competing risks: consistency and rates of convergence of the MLE. (English) Zbl 1360.62123

Summary: 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 N. P. Jewell et al. [Biometrika 90, No. 1, 183–197 (2003; Zbl 1034.62034)], 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 our paper [Ann. Stat. 36, No. 3, 1064–1089 (2008; Zbl 1216.62047)] to obtain the local limiting distributions of the estimators.

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62N01 Censored data models
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