## 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.

### MSC:

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

### Citations:

Zbl 0077.337; Zbl 0181.459
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