An Information-Theoretic Approach for Multivariate Skew-t Distributions and Applications

Salah Abid,Javier Contreras-Reyes,Uday Quaez

doi:10.3390/math9020146

Salah Abid, Javier Contreras-Reyes + Show 1 more

Open Access

https://doi.org/10.3390/math9020146

Copy DOI

Journal: Mathematics	Publication Date: Jan 11, 2021
Citations: 16	License type: CC BY 4.0

Affiliation: Mustansiriyah University, University of Valparaíso

Abstract

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented.

Highlights

Theoretic Approach for MultivariateFinite mixture models have used in the analysis of heterogeneous datasets due to its flexibility [1]
Azzalini and Dalla-Valle [9] and Azzalini and Capitanio [10] introduced the multivariate skew-normal and skew-t distributions as an alternative to multivariate normal distribution to deal with skewness and heavy-tailness in the data, respectively
Lin et al [11] proposed a development of finite mixture of skew-t (FMST) distributions, and more recently [4]

Summary

Introduction

Finite mixture models have used in the analysis of heterogeneous datasets due to its flexibility [1]. Considered bounds to approximates the Shannon entropy [8] Given that these applications have been developed in the normal mixture of densities context, several calculations of Shannon and Rényi entropies for non-normal distributions exist. Contreras-Reyes [13] presented the mutual information and Shannon entropy for multivariate skew-normal and skew-t distributions, respectively. Contreras-Reyes and Cortés [6] considered bounds to approximate the Rényi entropy for finite mixture of multivariate skew-normal (FMSN) distributions by using lower and upper. Simulation studies illustrate the behavior of Rényi entropy approximations for a given order α, skewness, and freedom degrees of the proposed mixture model.

Multivariate Skew-t Distribution

Entropies

Computational Implementation and Numerical Simulations

Application to Finite Mixtures of Multivariate Skew-t Distributions

Conclusions and Final Remarks