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Maximum likelihood estimation of a multi-dimensional log-concave density. Discussion of the paper by Cule, Samworth and Stewart

Maximum likelihood estimation of a multi-dimensional log-concave density. Discussion of the paper by Cule, Samworth and Stewart
Maximum likelihood estimation of a multi-dimensional log-concave density. Discussion of the paper by Cule, Samworth and Stewart
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 inline image 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 inline image in terms of a non‐differentiable convex optimization problem, and thus combine techniques of computational geometry with Shor's r‐algorithm to produce a sequence that converges to inline image. An R version of the algorithm is available in the package LogConcDEAD—log‐concave density estimation in arbitrary dimensions. We demonstrate that the estimator has attractive theoretical properties both when the true density is log‐concave and when this model is misspecified. For the moderate or large sample sizes in our simulations, inline image is shown to have smaller mean integrated squared error compared with kernel‐based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the expectation–maximization algorithm to fit finite mixtures of log‐concave densities.
1369-7412
589-590
Böhning, Dankmar
e59cede5-81be-4dfa-afec-0e81ac3b7f6c
Wang, Yong
a7913cc9-1250-4f71-ab1f-93480043bb6a
Böhning, Dankmar
e59cede5-81be-4dfa-afec-0e81ac3b7f6c
Wang, Yong
a7913cc9-1250-4f71-ab1f-93480043bb6a

Böhning, Dankmar and Wang, Yong (2010) Maximum likelihood estimation of a multi-dimensional log-concave density. Discussion of the paper by Cule, Samworth and Stewart. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72 (5), 589-590. (doi:10.1111/j.1467-9868.2010.00753.x).

Record type: Article

Abstract

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 inline image 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 inline image in terms of a non‐differentiable convex optimization problem, and thus combine techniques of computational geometry with Shor's r‐algorithm to produce a sequence that converges to inline image. An R version of the algorithm is available in the package LogConcDEAD—log‐concave density estimation in arbitrary dimensions. We demonstrate that the estimator has attractive theoretical properties both when the true density is log‐concave and when this model is misspecified. For the moderate or large sample sizes in our simulations, inline image is shown to have smaller mean integrated squared error compared with kernel‐based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the expectation–maximization algorithm to fit finite mixtures of log‐concave densities.

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Published date: 12 October 2010
Organisations: Statistics, Statistical Sciences Research Institute

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Local EPrints ID: 210473
URI: http://eprints.soton.ac.uk/id/eprint/210473
ISSN: 1369-7412
PURE UUID: 0108ac4f-33ab-42f9-a631-553e2e0f1044

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Date deposited: 09 Feb 2012 13:43
Last modified: 19 Dec 2018 17:30

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