Statistica Sinica 3(1993), 501-515

A PENALIZED MAXIMUM LIKELIHODD ESTIMATE OF

f(0+) WHEN f IS NON-INCREASING

Michael Woodroofe and Jiayang Sun

The University of Michigan

Abstract: The problem of estimating the value at 0+ of a non-increasing density f (on (0,∞) is considered. It is shown, by example, that the problem is interesting, and it is noted that the nonparametric maximum likelihood estimator is inconsistent. A penalized maximum likelihood estimator is derived as an alternative, and its properties studied through simulation and asymptotic analysis. In particular, the penalized maximum likelihood estimator is shown to be consistent.

Key words and phrases: Nonparametric maximum likelihood estimation, consistency, asymptotic distributions, simulation.