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Abstract: Let X 1,

., X n be independent and identically distributed random vectorswith a log-concave Lebesgue density f. We first prove that, with probabilityone, there exists a unique maximum likelihood estimator of f. The use of thisestimator is attractive because, unlike kernel density estimation, the methodis fully automatic, with no smoothing parameters to choose. Although theexistence proof is non-constructive, we are able to reformulate the issue ofcomputation in terms of a non-differentiable convex optimisation problem, andthus combine techniques of computational geometry with Shor-s r-algorithm toproduce a sequence that converges to the maximum likelihood estimate. For themoderate or large sample sizes in our simulations, the maximum likelihoodestimator is shown to provide an improvement in performance compared withkernel-based methods, even when we allow the use of a theoretical, optimalfixed bandwidth for the kernel estimator that would not be available inpractice. We also present a real data clustering example, which shows that ourmethodology can be used in conjunction with the Expectation-Maximisation EMalgorithm to fit finite mixtures of log-concave densities. An R version of thealgorithm is available in the package LogConcDEAD - Log-Concave DensityEstimation in Arbitrary Dimensions.

Author: Madeleine Cule, Richard Samworth, Michael Stewart



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