# Computing the maximum likelihood estimator of a multidimensional log-concave density

#### by Dr Richard Samworth

Institution: Cambridge
Date: Thu 26th July 2007
Time: 12:00 PM
Location: Room 213, Richard Berry Building, The University of Melbourne

Abstract: 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. The talk will be illustrated with pictures from the R package LogConcDEAD.

This is joint work with Madeleine Cule (Cambridge) and Michael Stewart (Sydney).