Limit distribution theory for maximum likelihood estimation of a log-concave density

TL;DR

This paper derives the limiting distributions for the nonparametric MLE of a log-concave density, including the mode estimator, and establishes its optimality through a new local asymptotic minimax lower bound.

Contribution

It provides the first detailed limiting distribution theory for the MLE of log-concave densities and introduces a new optimal mode estimator with proven convergence properties.

Findings

Limiting distributions depend on derivatives of the lower invelope process.

Mode estimator achieves optimal convergence rate.

New lower bound confirms estimator's optimality.

Abstract

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\varphi_0=\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.