Analytical properties of generalized Gaussian distributions

Alex Dytso, Ronit Bustin, H. Vincent Poor, Shlomo Shamai

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

The family of Generalized Gaussian (GG) distributions has received considerable attention from the engineering community, due to the flexible parametric form of its probability density function, in modeling many physical phenomena. However, very little is known about the analytical properties of this family of distributions, and the aim of this work is to fill this gap. Roughly, this work consists of four parts. The first part of the paper analyzes properties of moments, absolute moments, the Mellin transform, and the cumulative distribution function. For example, it is shown that the family of GG distributions has a natural order with respect to second-order stochastic dominance. The second part of the paper studies product decompositions of GG random variables. In particular, it is shown that a GG random variable can be decomposed into a product of a GG random variable (of a different order) and an independent positive random variable. The properties of this decomposition are carefully examined. The third part of the paper examines properties of the characteristic function of the GG distribution. For example, the distribution of the zeros of the characteristic function is analyzed. Moreover, asymptotically tight bounds on the characteristic function are derived that give an exact tail behavior of the characteristic function. Finally, a complete characterization of conditions under which GG random variables are infinitely divisible and self-decomposable is given. The fourth part of the paper concludes this work by summarizing a number of important open questions.

Original languageEnglish (US)
Article number6
JournalJournal of Statistical Distributions and Applications
Volume5
Issue number1
DOIs
StatePublished - Dec 1 2018

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Computer Science Applications
  • Statistics, Probability and Uncertainty

Keywords

  • Characteristic function
  • Generalized Gaussian distribution
  • Infinite divisibility
  • Mellin transform
  • Self-decomposition

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