# 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}
}
• Published 24 April 2008
• Mathematics
• Journal of the Royal Statistical Society: Series B (Statistical Methodology)
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…
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