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Informatika i Ee Primeneniya [Informatics and its Applications], 2020, Volume 14, Issue 4, Pages 33–36
DOI: https://doi.org/10.14357/19922264200405 (Mi ia694) 		 
On the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements. II

V. Yu. Korolevab

a Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russian Federation
b Institute of Informatics Problems, Federal Research Center “Computer Sciences and Control” of the Russian Academy of Sciences; 44-2 Vavilov Str., Moscow 119133, Russian Federation
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DOI: https://doi.org/10.14357/19922264200405
Abstract: The problem of description of the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements is considered. Unlike other versions of this problem in which the number of summed extreme order statistics and the threshold are fixed, here the specified threshold can be exceeded by an unpredictable number of sample elements. The situation is considered where the threshold infinitely increases as the sample size grows. It is demonstrated that in this case, the distribution of the ratio mentioned above can be approximated by the compound Poisson distribution in which the compounding law is the generalized Pareto distribution.
Keywords: sum of independent random variables, random sum, binomial distribution, Poisson approximation, extreme order statistic, Balkema – De Haan – Pickands theorem, generalized Pareto distribution, compound Poisson distribution.
Funding agency Grant number
Russian Foundation for Basic Research 19-07-00914
Moscow Center of Fundamental and Applied Mathematics
The research was supported by the Russian Foundation for Basic Research (project 19-07-00914). The research was conducted in accordance with the program of the Moscow Center for Fundamental and Applied Mathematics.
Received: 28.11.2019
Document Type: Article
Language: Russian
Citation: V. Yu. Korolev, “On the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements. II”, Inform. Primen., 14:4 (2020), 33–36
Citation in format AMSBIB
\Bibitem{Kor20}
\by V.~Yu.~Korolev
\paper On the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements.~II
\jour Inform. Primen.
\yr 2020
\vol 14
\issue 4
\pages 33--36
\mathnet{http://mi.mathnet.ru/ia694}
\crossref{https://doi.org/10.14357/19922264200405}
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