Résumé (EN)

We study the asymptotic behavior of the Maximum Likelihood estimator (MLE) of a density $f_0 = \exp \varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. Existence, form, characterizations and uniform rate of convergence of this so-called log--concave density estimator are given in \cite{rufibach_06_diss} and \cite{duembgen_rufibach_06}. It turns out that the problem of identifying the limiting distribution of the estimator is connected to that of the least squares estimator of a convex density on $[0, \infty)$, since in both estimation problems a specific characterization of the estimators in terms of distribution functions is (up to sign) the same. We find that the limiting local behavior depends on the \corr{second and third derivatives} at $0$ of $H_k$, the outer envelope of the integrated Brownian motion process plus a drift term depending on the number of vanishing derivatives of the true log--concave density at the estimation point. Furthermore, we establish the limiting distribution for the weak convergence of the first location of the maximum of the MLE to the true mode. Numerical simulations using the R--package \cite{logcondens} were performed to generate samples from the limiting distributions and calculate estimates of their extreme quantiles.