Statistica Sinica 23 (2013),

SMOOTHED LOG-CONCAVE MAXIMUM LIKELIHOOD

ESTIMATION WITH APPLICATIONS

Yining Chen and Richard J. Samworth

University of Cambridge

Abstract: We study the smoothed log-concave maximum likelihood estimator of a probability distribution on ${\Bbb R}^d$. This is a fully automatic nonparametric density estimator, obtained as a canonical smoothing of the log-concave maximum likelihood estimator. We demonstrate its attractive features both through an analysis of its theoretical properties and a simulation study. Moreover, we use our methodology to develop a new test of log-concavity, and show how the estimator can be used as an intermediate stage of more involved procedures, such as constructing a classifier or estimating a functional of the density. Here again, the use of these procedures can be justified both on theoretical grounds and through its finite sample performance, and we illustrate its use in a breast cancer diagnosis (classification) problem.

Key words and phrases: Classification, functional estimation, log-concave maximum likelihood estimation, smoothing, testing log-concavity.