On $f$-divergences between Cauchy distributions

Abstract

We prove that the $f$-divergences between univariate Cauchy distributions are all symmetric, and can be expressed as strictly increasing scalar functions of the symmetric chi-squared divergence. We report the corresponding scalar functions for the total variation distance, the Kullback-Leibler divergence, the squared Hellinger divergence, and the Jensen-Shannon divergence among others. Next, we give conditions to expand the $f$-divergences as converging infinite series of higher-order power chi divergences, and illustrate the criterion for converging Taylor series expressing the $f$-divergences between Cauchy distributions. We then show that the symmetric property of $f$-divergences holds for multivariate location-scale families with prescribed matrix scales provided that the standard density is even which includes the cases of the multivariate normal and Cauchy families. However, the $f$-divergences between multivariate Cauchy densities with different scale matrices are shown asymmetric. Finally, we present several metrizations of $f$-divergences between univariate Cauchy distributions and further report geometric embedding properties of the Kullback-Leibler divergence.

64 pages, 1 figure, 1 table

References in corpus (3)

• An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion
• Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution
• On information projections between multivariate elliptical and location-scale families

Cited by in corpus (3)

• Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models
• Information measures and geometry of the hyperbolic exponential families of Poincaré and hyperboloid distributions
• A note on the $f$-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters