We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp ϕ0 where ϕ0 is a concave function on ℝ. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0=log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) 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.
"Limit distribution theory for maximum likelihood estimation of a log-concave density." Ann. Statist. 37 (3) 1299 - 1331, June 2009. https://doi.org/10.1214/08-AOS609