Maximum likelihood estimation of a multi-dimensional log-concave density
Let "X"<sub>1</sub>,…,"X"<sub>"n"</sub> 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 <formula format="inline"><file name="rssb_753_mu1.gif" type="gif" /></formula> 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 <formula format="inline"><file name="rssb_753_mu2.gif" type="gif" /></formula> 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 <formula format="inline"><file name="rssb_753_mu3.gif" type="gif" /></formula>. 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, <formula format="inline"><file name="rssb_753_mu4.gif" type="gif" /></formula> 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. Copyright (c) 2010 Royal Statistical Society.
Year of publication: |
2010
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Authors: | Cule, Madeleine ; Samworth, Richard ; Stewart, Michael |
Published in: |
Journal of the Royal Statistical Society Series B. - Royal Statistical Society - RSS, ISSN 1369-7412. - Vol. 72.2010, 5, p. 545-607
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Publisher: |
Royal Statistical Society - RSS |
Saved in:
freely available
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