Kullback–Leibler Divergence Measure for Multivariate Skew-Normal Distributions

Abstract

The aim of this work is to provide the tools to compute the well-known Kullback-Leibler divergence measure for the flexible family of multivariate skew-normal distributions. In particular, we use the Jeffreys divergence measure to compare the multivariate normal distribution with the skew-multivariate normal distribution, showing that this is equivalent to comparing univariate versions of these distributions. Finally, we applied our results on a seismological catalogue data set related to the 2010 Maule earthquake. Specifically, we compare the distributions of the local magnitudes of the regions formed by the aftershocks.

Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus)

“…Inserting the ML estimation (fixed) parameters represents the simplest evaluation of these bounds [22]. However, between the lower and upper Rényi entropy bounds exists a considerable distance.…”

“…For the case m = 1, Contreras-Reyes and Arellano-Valle [22] consider the upper bound of the property (i) of Proposition 2 to approximate the Shannon entropy of an SN distribution using the property (ii) of Proposition 2. In this Proposition 2(ii), the left side includes an integral related to a product of two skew-normal densities.…”

Generalized Skew-Normal Negentropy and Its Application to Fish Condition Factor Time Series

“…Contreras-Reyes and Arellano-Valle [6] considered the result of Kupperman [30] to develop an asymptotic test of complete homogeneity in terms of the J divergence between two SN distributions. The SN distribution satisfies all the aforementioned regularity conditions when skewness parameter η = 0.…”

“…In both cases, we compare the SE and negentropies obtained from their series expansions with their corresponding "exact" versions computed from the Quadpack numerical integration method of Piessens et al [31]. More precisely, the "exact" expected values E{ζ 0 (τZ τ )} and E{ζ 0 (τZ * τ )} are computed using the Quadpack method as in Arellano-Valle et al [16], Contreras-Reyes and Arellano-Valle [6] or Contreras-Reyes [18]. From the series expansions, the SE and negentropies were carried out for k = 12 as in Withers and Nadarajah [19].…”

“…Stehlík [3] proved results on the decomposition of Kullback-Leibler (KL) divergences [4] in the gamma and normal family for divergence between the Maximum Likelihood Estimator (MLE) of the canonical parameter and the canonical parameter of the regular exponential family [5]. Contreras-Reyes and Arellano-Valle [6] considered Jeffrey's (J) divergence [7] to compare the multivariate Skew-Normal (SN) from the normal distribution, and Gómez-Villegas et al [8] assessed the effect of kurtosis deviations from normality on conditional distributions, such as the multivariate exponential power family. Main et al [9] evaluated the local effect of asymmetry deviations from normality using the KL divergence measure of the SN distribution and then compared the local sensitivity with Mardia's and Malkovich-Afifi's skewness indexes.…”