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

Irina Shevtsova and Mikhail Tselishchev
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Irina Shevtsova: Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Mikhail Tselishchev: Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia

Mathematics, 2021, vol. 9, issue 13, 1-8

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.

Keywords: Rényi theorem; law of large numbers; Kantorovich distance; Kolmogorov metric; zeta-structured metrics; geometric random sum; generalized negative binomial random sum; exponential distribution; generalized gamma distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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