ESTIMATING A MONOTONE DENSITY FROM CENSORED OBSERVATIONS

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 der ivative of the least concave majorant of the empirical distribution fu nction, and its local and global behavior was investigated, respective ly, by Prakasa Rao and Groeneboom. In this paper, we present a necessa ry and sufficient condition, a self-consistency equation and an analyt ic solution for the NPMLE, and we extend Prakasa Rao's result to the c ensored model.