Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics

Victor Korolev

doi:10.3390/math8111918

Abstract

In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are proved. Unlike the conventional analytical approach, here the presentation follows the lines of a kind of arithmetical approach in the space of random variables or vectors. Here the operation of scale mixing in the space of distributions is replaced with the operation of multiplication in the space of random vectors/variables under the assumption that the multipliers are independent. By doing so, the reasoning becomes much simpler, the proofs become shorter and some general features of the distributions under consideration become more vivid. The first part of the paper concerns the univariate case. Some known results are discussed and simple alternative proofs for some of them are presented as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions. The results obtained here open the way to deal with natural multivariate analogs of EP distributions. In the second part of the paper, we discuss the conventionally defined multivariate EP distributions and introduce the notion of projective EP (PEP) distributions. The properties of multivariate EP and PEP distributions are considered as well as limit theorems establishing the conditions for the convergence of multivariate statistics constructed from samples with random sizes (including random sums of random vectors) to multivariate elliptically contoured EP and projective EP laws. The results obtained here give additional theoretical grounds for the applicability of EP and PEP distributions as asymptotic approximations for the statistical regularities observed in data in many fields.

Highlights

In this paper we prove more general results than those presented in [20] and demonstrate that the exponential power (EP) distribution can be asymptotic in simple limit theorems for those statistics constructed from samples with random sizes, that are asymptotically normal, if the sample size is non-random, in particular, in the scheme of random summation
The representation of an EP distribution with 0 < α 6 2 as a normal scale mixture can be proved using the famous ‘multiplication’ Theorem 3.3.1 in [21]. In these representations the mixing distributions are defined via stable densities. We show that these mixing laws that play an auxiliary role in the theory of EP distributions, can play quite a separate role being limit laws for the random sample size providing that the extreme order statistics follow the gamma distribution
We give a survey of main results concerning univariate and multivariate EP distributions, consider the properties of mixing distributions appearing in the generalizations mentioned above and prove some asymptotic results based on mixture representations for EP and related distributions

Summary

Introduction

We give a survey of main results concerning univariate and multivariate EP distributions, consider the properties of mixing distributions appearing in the generalizations mentioned above and prove some asymptotic results based on mixture representations for EP and related distributions. We discuss some known results mentioned in [14] and present simple alternative proofs of some of them as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions.

Mathematical Preliminaries d

Higher-Order EP Scale Mixture Representations and Related Topics

Extensions of Gleser’s Theorem for Gamma Distributions

Alternative Mixture Representations

Some Limit Theorems for Extreme Order Statistics in Samples with Random Sizes

Conventional Approach Higher-Order EP Scale Mixture Representation

Multivariate Projective Exponential Power Distributions