# Maximum likelihood estimation of a multi‐dimensional log‐concave density

@article{Cule2008MaximumLE, title={Maximum likelihood estimation of a multi‐dimensional log‐concave density}, author={Madeleine L. Cule and Richard J. Samworth and Michael I. Stewart}, journal={Journal of the Royal Statistical Society: Series B (Statistical Methodology)}, year={2008}, volume={72} }

Summary. Let X1,…,Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log‐concave maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non‐constructive, we can reformulate the issue of computing in terms of a non‐differentiable…

## 190 Citations

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An algorithm using projected stochastic gradient method is proposed, in which biased gradient estimates are obtained through an Markov chain Monte Carlo sampling procedure that is efficient for log-concave densities.

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- Computer Science, MathematicsArXiv
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This work studies the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities and presents the first computationally efficient algorithm for this problem, based on a natural convex optimization formulation of the underlying problem and a projected stochastic subgradient method.

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