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Approximation by Log-Concave Distributions

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Multivariate distributions with log-concave densities are an interesting nonparametric model containing many standard parametric families such as Gaussian distributions. In the first part of the talk it is shown that this nonparametric model behaves almost like a parametric model in a certain sense. Moreover, we discuss the approximation of an arbitrary distribution P by a log-concave one with respect to a Kullback-Leibler type distance. Necessary and sufficient conditions for the existence of such an approximation are presented. Furthermore, this approximation depends continuously on the distribution P with respect to Wasserstein distance which has direct implications for nonparametric maximum likelihood estimation. In the second part we present applications to quantile estimation in non- and semiparametric regression models. ... This is based on joint work with Andre Huesler, Kaspar Rufibach, Richard Samworth and Dominic Schuhmacher

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