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

Irina Shevtsova,Mikhail Tselishchev

doi: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.

Highlights

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 the zeta-structured estimates of the rate of convergence in the Rényi theorem
Negative binomial and generalized negative binomial random sums naturally arise in various applications, such as meteorology, insurance, and financial mathematics
In [1], it was demonstrated that the generalized negative binomial distribution has an excellent concordance with the empirical distribution of the duration of wet periods measured in days in Potsdam and Elista