## The Annals of Statistics

### Estimating a Monotone Density from Censored Observations

#### Abstract

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

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

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.aos/1176325628

Digital Object Identifier
doi:10.1214/aos/1176325628

Mathematical Reviews number (MathSciNet)
MR1311975

Zentralblatt MATH identifier
0821.62016

JSTOR