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

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form $f_0 = \exp \phi_0$ where $\phi_0$ is a concave function on $\mathbb{R}$. Existence, form, characterizations and uniform rates of convergence of the MLE are given by Rufibach (2006) and D\"umbgen and Rufibach (2007). The characterization of the log-concave MLE in terms of distribution functions is the same (up to sign) as the characterization of the least squares estimator of a convex density on $[0,\infty)$ as studied by Groeneboom, Jongbloed, and Wellner (2001b). We use this connection to show that the limiting distributions of the MLE and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem. In particular, changing the smoothness assumptions of Groeneboom, Jongbloed, and Wellner (2001b) slightly by allowing some higher derivatives to vanish at the point of interest, we find that 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.