  We first 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 $f$-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 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.
