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

Victor Korolev and Alexander Zeifman ()
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Victor Korolev: Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, Moscow 119899, Russia
Alexander Zeifman: Federal Research Center “Computer Sciences and Control” of the Russian Academy of Sciences, 44-2 Vavilova St., Moscow 119333, Russia

Mathematics, 2023, vol. 11, issue 18, 1-25

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.

Keywords: generalized Student distribution; generalized Burr distribution; exponential power distribution; uniform distribution; Snedecor–Fisher distribution; gamma-distribution; scale mixture; multiplication theorem; limit theorem; random sum; extreme order statistic (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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