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Analytical properties of generalized Gaussian distributions

Author

Listed:
  • Alex Dytso

    (Department of Electrical Engineering, Princeton University)

  • Ronit Bustin

    (Department of Electrical Engineering, Technion-Israel Institute of Technology)

  • H. Vincent Poor

    (Department of Electrical Engineering, Princeton University)

  • Shlomo Shamai

    (Department of Electrical Engineering, Technion-Israel Institute of Technology)

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.

Suggested Citation

  • Alex Dytso & Ronit Bustin & H. Vincent Poor & Shlomo Shamai, 2018. "Analytical properties of generalized Gaussian distributions," Journal of Statistical Distributions and Applications, Springer, vol. 5(1), pages 1-40, December.
    • Handle: RePEc:spr:jstada:v:5:y:2018:i:1:d:10.1186_s40488-018-0088-5
    DOI: 10.1186/s40488-018-0088-5
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    References listed on IDEAS

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    1. Wolf-Dieter Richter, 2016. "Exact inference on scaling parameters in norm and antinorm contoured sample distributions," Journal of Statistical Distributions and Applications, Springer, vol. 3(1), pages 1-16, December.
    2. Goodman, Irwin R. & Kotz, Samuel, 1973. "Multivariate [theta]-generalized normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 204-219, June.
    3. Haim Levy, 1992. "Stochastic Dominance and Expected Utility: Survey and Analysis," Management Science, INFORMS, vol. 38(4), pages 555-593, April.
    4. Saralees Nadarajah, 2005. "A generalized normal distribution," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(7), pages 685-694.
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    Cited by:

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    2. Victor Korolev, 2020. "Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics," Mathematics, MDPI, vol. 8(11), pages 1-27, November.
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    4. Wolf-Dieter Richter, 2019. "On (p1,…,pk)-spherical distributions," Journal of Statistical Distributions and Applications, Springer, vol. 6(1), pages 1-18, December.

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