On the tails of log-concave density estimators

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 ${\hat{\phi }}_n=\log {\hat{f}}_n$, 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 ${\hat{f}}_n,{\hat{\phi }}_n$ and ${\hat{\phi }}_n^{\prime }$ 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.