Multivariate $\alpha$-normal distributions

TL;DR

This paper introduces the multivariate alpha-normal distribution, a new family derived from symmetrized power transformations of the normal distribution, and explores its properties, moments, entropy, and limiting behavior.

Contribution

It defines the multivariate alpha-normal distribution with alpha-normal marginals and analyzes its properties, moments, entropy, and limiting distribution as alpha approaches infinity.

Findings

Derived formulas for moments and differential entropy.

Defined the joint distribution as a meta-Gaussian with alpha-normal marginals.

Investigated the limiting distribution as alpha tends to infinity.

Abstract

The Weibull distribution can be obtained using a power transformation from the standard exponential distribution. In this article, we will consider a symmetrized power transformation of a random variable with the standard normal distribution. We will call its distribution the $\alpha$-{\it normal (Gaussian) distribution}. We examine properties of this distribution in detail. We calculate moments and consider the moment problem of $\alpha$-normal distribution. We derive the formula of its differential entropy and (exponential) Orlicz norm. % of $\alpha$-normal random variables. Moreover, we define the joint distribution function of the multivariate $\alpha$-normal distribution as a meta-Gaussian distribution with $\alpha$-normal marginals. We consider also the limiting distribution as $\alpha$ tends to infinity.