Global rates of convergence of the MLEs of log-concave and $s$-concave densities
Charles R. Doss, Jon A. Wellner
The Annals of Statistics · 2016 · Volume 44 · Issue 3
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 $s<-1$.
Keywords
$s$-concave, 62G05, 62G07, 62G20, Bracketing entropy, consistency, Empirical processes, global rate, Hellinger metric, log-concave
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