On the tails of log-concave density estimators

TL;DR

This paper demonstrates that the nonparametric maximum likelihood estimator for univariate log-concave densities has strong consistency properties in the tail regions, with errors diminishing uniformly on specified sequences.

Contribution

It establishes uniform convergence of the estimated log-concave density and its derivatives in tail regions, using novel inequalities for truncated moments of log-concave distributions.

Findings

Uniform convergence of density estimates in tails

Error bounds for estimated log-densities and derivatives

Development of inequalities for truncated moments

Abstract

It is shown that the nonparametric maximum likelihood estimator of a univariate log-concave probability density satisfies desirable consistency properties in the tail regions. Specifically, let $P$ and $f$ denote the true underlying distribution and density, respectively. If $\hat{f}_n$ is the estimated log-concave density, and $\hat{\varphi}_n = \log \hat{f}_n$, then we specify sequences $(b_n)_{n\in \mathbb{N}}$ such that $P([b_n,\infty)) \to 0$ at a specific speed, ensuring that the absolute errors or absolute relative errors of $\hat{f}_n, \ \hat{\varphi}_n$ and $\hat{\varphi}_n'$ converge to zero uniformly on sets $[a, b_n]$. The main tools, besides characterizations of $\hat{f}_n$, are exponential and maximal inequalities for truncated moments of log-concave distributions, which are of independent interest.