Maximum likelihood estimation of a multidimensional log-concave density

@article{Cule2008MaximumLE,
  title={Maximum likelihood estimation of a multidimensional log-concave density},
  author={Madeleine L. Cule and Richard J. Samworth and Michael I. Stewart},
  journal={arXiv: Methodology},
  year={2008}
}
Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique 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 are able to reformulate the issue of computation in terms of a non… 
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We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in R d . Our study covers both the case where
Maximum likelihood estimation of a multivariate log-concave density
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This paper proposes an alternative approach using maximum likelihood under a qualitative assumption on the shape of the density, specifically log-concavity, and introduces a method for assessing the suitability of this shape constraint and applies it to several simulated datasets and one real dataset.
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Discussion of the paper by Cule, Samworth and Stewart: \Maximum likelihood estimation of a multidimensional log-concave density"
White et al. (2010) consider an exact method of sampling-based Bayesian inference in the context of stochastic population models. This gives rise to a posterior distribution of the parameters of the
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