Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

TL;DR

This paper investigates the properties and convergence rates of nonparametric maximum likelihood estimators for log-concave densities, distribution functions, and hazard functions, providing theoretical guarantees and rates of convergence.

Contribution

It establishes fundamental properties and convergence rates of the MLE for log-concave densities and related functions, advancing understanding of their statistical behavior.

Findings

Convergence rate for density and hazard estimator is at least (log(n)/n)^{1/3}

Distribution function estimator converges at a rate o_p(n^{-1/2})

Provides characterizations and properties of the estimators under regularity conditions

Abstract

We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least $(\log(n)/n)^{1/3}$ and typically $(\log(n)/n)^{2/5}$, whereas the difference between the empirical and estimated distribution function vanishes with rate $o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions.