Asymptotic behaviour of insurance company reserve

V. E. Bening, R. V. Pirogov


The paper presents statistical modeling of some random sum models. Poisson-binomial, Poisson, geometric random sums and random sums with volume of three-point symmetric distribution were modeled and analyzed. The results of this work are relevant not only for insurance business, but also for any other business that involves random accumulation of outcomes. From the nature of distribution of random volume will record the limit distribution of the random amount and the speed of its convergence. Structure of the limit distribution plays an important role in the evaluation of quintiles and testing of hypotheses. The necessary statistical modeling is carried out, which confirms the practical application of asymptotic models. Difference of limit distributions in the class of mixed Poisson sums is underlined. Estimates of the convergence rate of some simulated random sums to their limit distributions are made. To estimate Poisson-binomial, Poisson and mixed Poisson random sums, Berry-Esseen inequality was applied. To estimate quantile of random sum with volume of threepoint symmetric distribution, an asymptotic decomposition of distribution function of non-normalized statistics of random sum was carried out. Calculation of the required reserve of insurance company is to estimate the level quintile close to one.

Full Text:

PDF (Russian)


Pirogov R.V. Asymptotic approximation of statistics based on the

sample of negative binomial distribution – 2019: International Journal

of Open Information Technologies.

Berry A. C. The accuracy of the Gaussian approximation to the sum

of independent variates // Trans. Amer. Math. Soc. – 1941. – Vol. 49.

– P. 122-136.

SHevtsova I.G. Estimates of the accuracy of asymmetric probabilistic

models 2018.

Johnson, N.L., Kemp, A.W., and Kotz, S. Univariate Discrete

Distributions, 3rd Edition, Wiley – 2005.

Huiming, Zhang; Yunxiao Liu; Bo Li. ”Notes on discrete compound

Poisson model with applications to risk theory”. Insurance:

Mathematics and Economics. – 2014.

Bening V., Galiyeva N., Korolev V. On concentration functions

of regular statistics constructed from samples with random size//

XXX International Seminar on Stability Problems for Stochastic

Models and VI InternationalWorkshop «Applied Problems in Theory

of Probfbilities and Mathematical Statistics Related to Modeling of

Information Systems» – 2012. – С.15-18.

SHevtsova I.G. Absolute constants in the Berry-Esseen inequality and

its structural and non-uniform clarifications – 2013. – vol. 7, no. 1.

– p. 124–125.

SHevtsova I.G. Absolute constants in inequalities like Berry-Esseen –

– vol. 456, no. 6. – p. 650–654.

Grandell J. Mixed Poisson processes. – London : Chapman and Hall,

Bening V. E., Korolev V. Y. Generalized Poisson Models and their

Applications in Insurance and Finance. – Utrecht, The Netherlands :

VSP, 2002.

Korolev V. Y. A general theorem on the limit behavior of

superpositions of independent random processes with applications to

Cox processes. Journal of Mathematical Sciences // J. Math. Sci. –

– Vol. 81, no. 5. – P. 2951-2956.

Gavrilichenko S.V., Korolev V.U. Estimates of the convergence rate

of mixed Poisson random sums // Systems and tools of informatics.

Special edition. – 2006. – p. 248-257.

Petrov V.V. Sums of independent random variables. – 1972.

Bening V. E. On the asymptotic behavior of the deficiency of some

statistical estimators based on samples with random sizes. – 1972. RDocumentation 2018.


  • There are currently no refbacks.

Abava  Absolutech Convergent 2020

ISSN: 2307-8162