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Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data. (English) Zbl 1333.62129

Summary: The second and the third author [J. Multivariate Anal. 69, No. 2, 155–166 (1999; Zbl 0931.62084)] discussed generalized maximum likelihood estimation of the joint distribution function of a multivariate random vector whose coordinates are subject to interval censoring. They established uniform consistency of the generalized MLE (GMLE) of the distribution function under the assumption that the random vector is independent of the censoring vector and that both of the vector distributions are discrete. We relax these assumptions and establish consistency results of the GMLE under a multivariate mixed case interval censorship model. A. van der Vaart and J. A. Wellner [Prog. Probab. 47, 115–133 (2000; Zbl 0967.60037)] and the first author [Consistency of the generalized MLE with multivariate mixed case interval-censored data. Binghamton, NY: Binghamton University (PhD Thesis) (2000)] independently proved strong consistency of the GMLE in the \(L_{1}(\mu)\)-topology, where \(\mu\) is a measure derived from the joint distribution of the censoring variables. We establish strong consistency of the GMLE in the topologies of weak convergence and pointwise convergence, and eventually uniform convergence under appropriate distributional assumptions and regularity conditions.

MSC:

62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
62N01 Censored data models
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