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

Authors :
Nielsen, Frank
Okamura, Kazuki
Source :
Statistics and Computing , Volume 34, article number 60, (2024)
Publication Year :
2022

Abstract

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.<br />Comment: 17 pages, 1 table, 1 figure

Details

Database :
arXiv
Journal :
Statistics and Computing , Volume 34, article number 60, (2024)
Publication Type :
Report
Accession number :
edsarx.2204.10952
Document Type :
Working Paper
Full Text :
https://doi.org/10.1007/s11222-023-10373-6