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 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 H_k, 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.