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

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

2007-08-24

arxiv.org/abs/0708.3400v3

Annals of Statistics 2009, Vol. 37, No. 3, 1299-1331

Mathematics

Statistics Theory

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

Scientific paper

10.1214/08-AOS609

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.

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