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

Abstract

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp {\phi }_0$ where ${\phi }_0$ 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 ${\phi }_0=\log f_0$ 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.

Published in at http://dx.doi.org/10.1214/08-AOS609 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Chernoff's density is log-concave
• Estimation of a discrete probability under constraint of k-monotony
• A Bayesian nonparametric approach to log-concave density estimation
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• Semiparametric Time Series Models with Log-concave Innovations: Maximum Likelihood Estimation and its Consistency