On the f-divergences between densities of a multivariate location or scale family

In this paper, we first extend the result of Ali and Silvey [J R Stat Soc Ser B, 28:131–142, 1966] who proved that any f-divergence between two isotropic multivariate Gaussian distributions amounts to a corresponding strictly increasing scalar function of their corresponding Mahalanobis distance. We give sufficient conditions on the standard probability density function generating a multivariate location family and the function generator f in order to generalize this result. This property is useful in practice as it allows to compare exactly f-divergences between densities of these location families via their corresponding Mahalanobis distances, even when the f-divergences are not available in closed-form as it is the case, for example, for the Jensen–Shannon divergence or the total variation distance between densities of a normal location family. Second, we consider f-divergences between densities of multivariate scale families: We recall Ali and Silvey ’s result that for normal scale families we get matrix spectral divergences, and we extend this result to densities of a scale family.