Maximum likelihood estimation of a multidimensional log-concave density

TL;DR

This paper introduces a unique, fully automatic maximum likelihood estimator for multidimensional log-concave densities, demonstrating its theoretical existence, computational approach, and superior performance over kernel methods in simulations and real data clustering.

Contribution

It proves the almost sure existence and uniqueness of the MLE for log-concave densities and develops a computational method combining geometry and convex optimization techniques.

Findings

MLE exists and is unique with probability one

Estimator outperforms kernel density methods in simulations

Applicable to mixture models with EM algorithm

Abstract

Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we are able to reformulate the issue of computation in terms of a non-differentiable convex optimisation problem, and thus combine techniques of computational geometry with Shor's r-algorithm to produce a sequence that converges to the maximum likelihood estimate. For the moderate or large sample sizes in our simulations, the maximum likelihood estimator is shown to provide an improvement in performance compared with kernel-based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the Expectation--Maximisation (EM) algorithm to fit finite mixtures of log-concave densities. An R version of the algorithm is available in the package LogConcDEAD -- Log-Concave Density Estimation in Arbitrary Dimensions.