On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

Irina Shevtsova; Mikhail Tselishchev

doi:10.3390/math9131571

journal article Open Access Jul 04, 2021

On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

Irina Shevtsova Mikhail Tselishchev

Mathematics Vol. 9 No. 13 pp. 1571 · MDPI AG

View at Publisher Save 10.3390/math9131571

Abstract

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

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Citations

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References

Details

Published: Jul 04, 2021
Vol/Issue: 9(13)
Pages: 1571
License: View

Authors

I

Irina Shevtsova

Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China; Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia; Federal Research Center ”Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia; Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia

M

Mikhail Tselishchev

Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia

Funding

Russian Foundation for Basic Research Award: 20-31-70054

Ministry For Science and Higher Education of Russia Award: MD-5748.2021.1.1

Cite This Article

Irina Shevtsova, Mikhail Tselishchev (2021). On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums. Mathematics, 9(13), 1571. https://doi.org/10.3390/math9131571

On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

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