Estimating a monotone density from censored observations. (English) Zbl 0821.62016

Summary: We study the nonparametric maximum likelihood estimator (NPMLE) for a concave distribution function \(F\) and its decreasing density \(f\) based on right-censored data. Without the concavity constraint, the NPMLE of \(F\) is the Kaplan-Meier product-limit estimator. If there is no censoring, the NPMLE of \(f\), derived by U. Grenander [Skand. Aktuarietidskr. 1956, 125-153 (1957; Zbl 0077.337)], is the left derivative of the least concave majorant of the empirical distribution function, and its local and global behavior was investigated, respectively, by B. L. S. Prakasa Rao [Sankyā, Ser. A 31, 23-36 (1969; Zbl 0181.459)] and P. Groeneboom [see “Estimating a monotone density.” Amsterdam: Report. Stichting Mathematisch Centrum (1984)]. We present a necessary and sufficient condition, a self-consistency equation and an analytic solution for the NPMLE, and we extend Prakasa Rao’s result to the censored model.


62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
62E20 Asymptotic distribution theory in statistics
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