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We present theoretical properties of the log-concave maximum likelihoodestimator of a density based on an independent and identically distributedsample in $\mathbb{R}^d$. Our study covers both the case where the trueunderlying density is log-concave, and where this model is misspecified. Webegin by showing that for a sequence of log-concave densities, convergence indistribution implies much stronger types of convergence -- in particular, itimplies convergence in Hellinger distance and even in certain exponentiallyweighted total variation norms. In our main result, we prove the existence anduniqueness of a log-concave density that minimises the Kullback--Leiblerdivergence from the true density over the class all log-concave densities, andalso show that the log-concave maximum likelihood estimator converges almostsurely in these exponentially weighted total variation norms to this minimiser.In the case of a correctly specified model, this demonstrates a strong type ofconsistency for the estimator; in a misspecified model, it shows that theestimator converges to the log-concave density that is closest in theKullback--Leibler sense to the true density.

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