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

22 April 2022
Frank Nielsen
K. Okamura
ArXiv (abs)PDFHTML
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 fff-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 fff in order to generalize this result. In that case, one can compare exactly fff-divergences between densities of these location families via their Mahalanobis distances. In particular, this proves useful when the fff-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 fff-divergences between these multivariate location-scale families amount equivalently to fff-divergences between corresponding univariate location-scale families. We present several applications of these results.

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