Global Rates of Convergence of the MLEs of Log-concave and s-concave Densities

TL;DR

This paper establishes the convergence rates of maximum likelihood estimators for log-concave and s-concave densities, showing a rate of no worse than n^{-2/5} in the Hellinger metric, and discusses existence issues.

Contribution

It provides the first global convergence rates for MLEs of s-concave densities and clarifies the existence conditions for these estimators.

Findings

Convergence rate of MLE in Hellinger metric is at most n^{-2/5} for -1<s<

MLE does not exist for s-concave densities with s < -1

Results unify understanding of MLE behavior across different s-concave classes

Abstract

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-2/5}$ when $-1 < s < \infty$ where $s=0$ corresponds to the log-concave case. We also show that the MLE does not exist for the classes of $s$-concave densities with $s < - 1$.