Dr Richard Samworth, Statistical Laboratory, Cambridge
Computing the maximum likelihood estimator of a multidimensional log-concave density

We show that if $X_1,...,X_n$ are a random sample from a log-concave density $f$ in $\mathbb{R}^d$, then with probability one there exists a unique maximum likelihood estimator $\hat{f}_n$ of $f$.  The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. The existence proof is non-constructive, however, and in practice we require an iterative algorithm that converges to the estimator.  By reformulating the problem as one of non-differentiable convex optimisation, we are able to exhibit such an algorithm.  We will also show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities.  The talk will be illustrated with pictures from the R package LogConcDEAD.

This is joint work with Madeleine Cule (Cambridge), Bobby Gramacy (Cambridge) and Michael Stewart (University of Sydney).