On the tails of log-concave density estimators

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.