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We first extend the result of Ali and Silvey [Journal of the RoyalStatistical Society: Series B, 28.1 (1966), 131-142] who first reported thatany $f$-divergence between two isotropic multivariate Gaussian distributionsamounts to a corresponding strictly increasing scalar function of theircorresponding Mahalanobis distance. We report sufficient conditions on thestandard probability density function generating a multivariate location familyand the function generator $f$ in order to generalize this result. Thisproperty is useful in practice as it allows to compare exactly $f$-divergencesbetween densities of these location families via their correspondingMahalanobis distances, even when the $f$-divergences are not available inclosed-form as it is the case, for example, for the Jensen-Shannon divergenceor the total variation distance between densities of a normal location family.Second, we consider $f$-divergences between densities of multivariate scalefamilies: We recall Ali and Silvey 's result that for normal scale families weget matrix spectral divergences, and we extend this result to densities of ascale family.

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