Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density

TL;DR

This paper investigates the theoretical properties of the log-concave maximum likelihood estimator in multidimensional density estimation, establishing convergence and consistency results under both correct and misspecified models.

Contribution

It proves the existence and uniqueness of the KL-minimising log-concave density and shows the estimator's convergence in strong norms, extending understanding of its theoretical behavior.

Findings

Convergence in distribution implies stronger convergence types.

Existence and uniqueness of the KL-minimising log-concave density.

Almost sure convergence of the MLE to the KL-minimiser.

Abstract

We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $\mathbb{R}^d$. Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much stronger types of convergence -- in particular, it implies convergence in Hellinger distance and even in certain exponentially weighted total variation norms. In our main result, we prove the existence and uniqueness of a log-concave density that minimises the Kullback--Leibler divergence from the true density over the class all log-concave densities, and also show that the log-concave maximum likelihood estimator converges almost surely in these exponentially weighted total variation norms to this minimiser. In the case of a correctly specified model, this demonstrates a strong type of consistency for the estimator; in a misspecified model, it shows that the estimator converges to the log-concave density that is closest in the Kullback--Leibler sense to the true density.