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Quasi-Exponentiated Normal Distributions: Mixture Representations and Asymmetrization

Author

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  • Victor Korolev

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, 119899 Moscow, Russia
    Federal Research Center “Computer Sciences and Control” of the Russian Academy of Sciences, 44-2 Vavilova Str., 119333 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia)

  • Alexander Zeifman

    (Federal Research Center “Computer Sciences and Control” of the Russian Academy of Sciences, 44-2 Vavilova Str., 119333 Moscow, Russia
    Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia
    Vologda Research Center of the Russian Academy of Sciences, 556A Gorky Str., 160014 Vologda, Russia)

Abstract

In the paper, quasi-exponentiated normal distributions are introduced for any real power (exponent) no less than two. With natural exponents, the quasi-exponentiated normal distributions coincide with the distributions of the corresponding powers of normal random variables with zero mean. Their representability as scale mixtures of normal and exponential distributions is proved. The mixing distributions are written out in the closed form. Two approaches to the construction of asymmetric quasi-exponentiated normal distributions are described. A limit theorem is proved for sums of a random number of independent random variables in which the asymmetric quasi-exponentiated normal distribution is the limit law.

Suggested Citation

  • Victor Korolev & Alexander Zeifman, 2023. "Quasi-Exponentiated Normal Distributions: Mixture Representations and Asymmetrization," Mathematics, MDPI, vol. 11(17), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3797-:d:1232798
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    References listed on IDEAS

    as
    1. Küchler, Uwe & Tappe, Stefan, 2008. "Bilateral gamma distributions and processes in financial mathematics," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 261-283, February.
    2. Dilip B. Madan & King Wang, 2020. "Additive Processes with Bilateral Gamma Marginals," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(3), pages 171-188, May.
    3. Zhu, Dongming & Zinde-Walsh, Victoria, 2009. "Properties and estimation of asymmetric exponential power distribution," Journal of Econometrics, Elsevier, vol. 148(1), pages 86-99, January.
    4. Victor Korolev, 2020. "Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics," Mathematics, MDPI, vol. 8(11), pages 1-27, November.
    5. Victor Korolev, 2023. "Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions," Mathematics, MDPI, vol. 11(13), pages 1-27, June.
    6. Küchler, Uwe & Tappe, Stefan, 2008. "On the shapes of bilateral Gamma densities," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2478-2484, October.
    Full references (including those not matched with items on IDEAS)

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