Analytical properties of generalized Gaussian distributions

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.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

“…Hence, multiplying (17) with (15) gives (11). □ Now, some of the main results of this paper can be obtained.…”

“…Recent studies about generalized gamma and Gaussian probability distributions or involving these probability distributions may, for example, be found in [14] [15]. In this study these distributions are generalized further.…”

“…In this study these distributions are generalized further. For instance, analytical properties regarding moments, characteristic functions of the generalized Gaussian distribution and their possible applications are very well documented in Dytso et al [15]. However, there is no a direct formula to evaluate the n th moments of the generalized Gaussian distribution.…”

Applicational Statistics in Data Science and Machine Learning

“…Gaussian Distribution Dytso et al [6] described in his paper that Gaussian distribution has been very much accepted by the engineering community because of its flexible parametric form. Gaussian distribution, also known as normal distribution, is the best example of randomness.…”

On (p1,…,pk)-spherical distributions

“…The univariate q-generalized normal distribution or q-power exponential distribution has been parameterized in different ways in the literature, for a recent survey see Dytso et al (2018). For different purposes, any of these parameterizations can be used to derive modified representations of the distributions considered in this paper.…”