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2010 Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density
Madeleine Cule, Richard Samworth
Electron. J. Statist. 4: 254-270 (2010). DOI: 10.1214/09-EJS505

Abstract

We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in ℝd. Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much stronger types of convergence – in particular, it implies convergence in Hellinger distance and even in certain exponentially weighted total variation norms. In our main result, we prove the existence and uniqueness of a log-concave density that minimises the Kullback–Leibler divergence from the true density over the class of all log-concave densities, and also show that the log-concave maximum likelihood estimator converges almost surely in these exponentially weighted total variation norms to this minimiser. In the case of a correctly specified model, this demonstrates a strong type of consistency for the estimator; in a misspecified model, it shows that the estimator converges to the log-concave density that is closest in the Kullback–Leibler sense to the true density.

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Madeleine Cule. Richard Samworth. "Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density." Electron. J. Statist. 4 254 - 270, 2010. https://doi.org/10.1214/09-EJS505

Information

Published: 2010
First available in Project Euclid: 17 February 2010

zbMATH: 1329.62183
MathSciNet: MR2645484
Digital Object Identifier: 10.1214/09-EJS505

Subjects:
Primary: 62G07 , 62G20

Keywords: consistency , Kullback–Leibler divergence , Log-concavity , maximum likelihood estimation , model misspecification

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

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