On information projections between multivariate elliptical and location-scale families
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We study information projections with respect to statistical f-divergences between any two location-scale families. We consider a multivariate generalization of the location-scale families which includes the elliptical and the spherical subfamilies. By using the action of the multivariate location-scale group, we show how to reduce the calculation of f-divergences between any two location-scale densities to canonical settings involving standard densities, and derive thereof fast Monte Carlo estimators of f-divergences with good properties. Finally, we prove that the minimum f-divergence between a prescribed density of a location-scale family and another location-scale family is independent of the prescribed location-scale parameter. We interpret geometrically this property.
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