Abstract

A density on R^d is said to be log-concave if its logarithm is a concave function, and the estimation of a unknown log-concave density based on i.i.d. observations represents a central problem in the area of non-parametric inference under shape constraints. In contrast to traditional smoothing techniques, the log-concave maximum likelihood estimator is a fully automatic estimator which does not require the choice of any tuning parameters and therefore has the potential to offer practitioners the best of the parametric and non-parametric worlds. I will discuss some recent theoretical results on the performance of this estimator, with a particular focus on its ability to adapt to structural features of the target density.