Smoothed log-concave maximum likelihood estimation with applications

TL;DR

This paper introduces a smoothed log-concave maximum likelihood estimator for probability distributions, offering a fully automatic nonparametric density estimation method with theoretical analysis, simulation validation, and practical applications like testing log-concavity and classification.

Contribution

It proposes a novel smoothed log-concave MLE, analyzes its theoretical properties, and demonstrates its utility in hypothesis testing and classification tasks.

Findings

The estimator is fully automatic and nonparametric.

It performs well in finite sample scenarios.

It can be used for log-concavity testing and classification.

Abstract

We study the smoothed log-concave maximum likelihood estimator of a probability distribution on $\mathbb{R}^d$. 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.