The Annals of Statistics

Estimating a Monotone Density from Censored Observations

Youping Huang and Cun-Hui Zhang

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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 product-limit estimator proposed by Kaplan and Meier. If there is no censoring, the NPMLE of $f$, derived by Grenander, is the left derivative of the least concave majorant of the empirical distribution function, and its local and global behavior was investigated, respectively, by Prakasa Rao and Groeneboom. In this paper, 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.

Article information

Ann. Statist., Volume 22, Number 3 (1994), 1256-1274.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G30: Order statistics; empirical distribution functions 62E20: Asymptotic distribution theory

Nonparametric maximum likelihood estimation censored data monotone density self-consistency least concave majorant product limit estimator


Huang, Youping; Zhang, Cun-Hui. Estimating a Monotone Density from Censored Observations. Ann. Statist. 22 (1994), no. 3, 1256--1274. doi:10.1214/aos/1176325628.

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