Electronic Journal of Statistics

Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density

Madeleine Cule and Richard Samworth

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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 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.

Article information

Electron. J. Statist. Volume 4 (2010), 254-270.

First available in Project Euclid: 17 February 2010

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties

Consistency log-concavity Kullback–Leibler divergence maximum likelihood estimation model misspecification


Cule, Madeleine; Samworth, Richard. Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Statist. 4 (2010), 254--270. doi:10.1214/09-EJS505. http://projecteuclid.org/euclid.ejs/1266416850.

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