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

Huang, Youping; Zhang, Cun-Hui

Ann. Stat. 22, No. 3, 1256-1274 (1994).

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

Keywords:

uniform consistency; Brownian motion; monotone density; nonparametric maximum likelihood estimator; concave distribution; right-censored data; Kaplan-Meier product-limit estimator; least concave majorant; empirical distribution; self-consistency equation; analytic solution

Citations:

Zbl 0077.337; Zbl 0181.459