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Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results

Author

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  • Victor Korolev

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

  • Alexander Zeifman

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

Abstract

In this paper, the representability of the generalized Student’s distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws.

Suggested Citation

  • Victor Korolev & Alexander Zeifman, 2023. "Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results," Mathematics, MDPI, vol. 11(18), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3892-:d:1238715
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    References listed on IDEAS

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    2. Choy, S.T. Boris & Chan, C.M., 2003. "Scale Mixtures Distributions in Insurance Applications," ASTIN Bulletin, Cambridge University Press, vol. 33(1), pages 93-104, May.
    3. Victor Korolev, 2023. "Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions," Mathematics, MDPI, vol. 11(13), pages 1-27, June.
    4. Zhu, Dongming & Galbraith, John W., 2010. "A generalized asymmetric Student-t distribution with application to financial econometrics," Journal of Econometrics, Elsevier, vol. 157(2), pages 297-305, August.
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    6. Satya D. Dubey, 1968. "A compound weibull distribution," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 15(2), pages 179-188, June.
    7. 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.
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