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Multivariate $α$-normal distributions

Krzysztof Zajkowski

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 $α$-{\it normal (Gaussian) distribution}. We examine properties of this distribution in detail. We calculate moments and consider the moment problem of $α$-normal distribution. We derive the formula of its differential entropy and (exponential) Orlicz norm. % of $α$-normal random variables. Moreover, we define the joint distribution function of the multivariate $α$-normal distribution as a meta-Gaussian distribution with $α$-normal marginals. We consider also the limiting distribution as $α$ tends to infinity.

Multivariate $α$-normal distributions

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 -{\it normal (Gaussian) distribution}. We examine properties of this distribution in detail. We calculate moments and consider the moment problem of -normal distribution. We derive the formula of its differential entropy and (exponential) Orlicz norm. % of -normal random variables. Moreover, we define the joint distribution function of the multivariate -normal distribution as a meta-Gaussian distribution with -normal marginals. We consider also the limiting distribution as tends to infinity.

Paper Structure

This paper contains 6 sections, 5 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $\alpha$-normal distribution
  3. Comparison with the Weibull distribution
  4. Orlicz norm of the $\alpha$-normal distribution
  5. Multivariate $\alpha$-normal distributions
  6. Limiting distribution

Key Result

Proposition 2.5

i) The distribution function (d.f.) $\Phi_\alpha$ of $G_\alpha$ is of the form where $\Phi$ is the standard normal distribution function. ii) The probability density function $\varphi_\alpha$ of $G_\alpha$ is of the form

Figures (1)

  • Figure 1: Density function $\varphi_{\alpha}$ depending on the value of parameter $\alpha$.

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 3.1
  • ...and 11 more