Renyi and Shannon Entropies of Finite Mixtures of Multivariate Skew t-distributions

TL;DR

This paper extends the concepts of Shannon and Renyi entropies to multivariate skew t-distributions and their mixtures, providing bounds, approximations, and real data illustrations for these entropy measures.

Contribution

It introduces new entropy bounds and approximation methods for finite mixtures of multivariate skew t-distributions, expanding previous work on t-distributions.

Findings

Derived upper and lower bounds for entropies of skew t-distribution mixtures

Provided asymptotic expressions for Renyi entropy

Illustrated entropy behavior with real data examples

Abstract

Shannon and Renyi entropies are quantitative measures of uncertainty in a data set. They are developed by Renyi in the context of entropy theory. These measures have been studied in the case of the multivariate t-distributions. We extend these tools to the class of multivariate skew t-distributions and then to more families of finite mixture of multivariate skew t distributions. In particular, by using generalized Holder inequality and some properties of multinomial theorem, we find upper and lower bounds of entropies for these families. An approximate value of these entropies can be calculated. In addition, an asymptotic expression for Renyi entropy is given by approximation and by using some inequalities and properties of Lp spaces. Finally, we give a real data examples to illustrate the behavior of entropy of the mixture model under consideration.