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

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 $f$-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-scale family and the generator $f$ in order to generalize this result. In that case, one can compare exactly $f$-divergences between densities of these location families via their Mahalanobis distances. In particular, this proves useful when the $f$-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 $f$-divergences between these multivariate location-scale families amount equivalently to $f$-divergences between corresponding univariate location-scale families. We present several applications of these results.