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A note on the ff-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters

Abstract

We extend the result of Ali and Silvey [Journal of the Royal Statistical Society: Series B, 28.1 (1966), 131-142] who first reported that any ff-divergence between two isotropic multivariate Gaussian distributions amounts to a corresponding strictly increasing scalar function of their corresponding Mahalanobis distance. We report sufficient conditions on the standard probability density function generating a multivariate location-scale family and the generator ff in order to generalize this result. In that case, one can compare exactly ff-divergences between densities of these location families via their Mahalanobis distances. In particular, this proves useful when the ff-divergences are not available in closed-form as it is the case for example for the Jensen-Shannon divergence between multivariate isotropic Gaussian distributions. Furthermore, we show that the ff-divergences between these multivariate location-scale families amount equivalently to ff-divergences between corresponding univariate location-scale families. We present several applications of these results.

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