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Approximation of sums by compound Poisson distributions with respect to stop-loss distances

Published online by Cambridge University Press:  01 July 2016

S. T. Rachev  and
L. Rüschendorf
S. T. Rachev*
Affiliation:
University of California, Santa Barbara
L. Rüschendorf*
Affiliation:
University of Munster
*
∗Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA. Supported by Deutsche Forschungsgemeinschaft Grant.
∗∗Postal address: Institut für Mathematische Statistik, Westfälische Wilhelms-Universität, Einsteinstrasse 62, 4400 Münster, W. Germany.
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Abstract

The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 – pi that the ith summand is zero is not too large.

Keywords

RISK THEORYINSURANCE MATHEMATICS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 2 , June 1990 , pp. 350 - 374
Copyright
Copyright © Applied Probability Trust 1990 

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References

[1] Arak, T. V. (1981) On the convergence rate in Kolmogorov's uniform limit theorem I and II. Theory Prob. Appl. 26, 219–239, 437-451.Google Scholar
[2] Arak, T. V. and Zaitsev, A. Yu. (1988) Uniform limit theorems for sums of independent random variables. Proc. Steklov Inst. Math. 174, American Mathematical Society.Google Scholar
[3] Deheuvels, P. and Pfeifer, D. (1986) A semigroup approach to Poisson approximation. Ann. Prob. 14, 663–678.Google Scholar
[4] Deheuvels, P. and Pfeifer, D. (1988) On a relationship between Uspensky's theorem and Poisson approximations. Ann. Inst. Statist. Math. 40, 671–681.Google Scholar
[5] Deheuvels, P., Karr, A., Pfeifer, D. and Serfling, R. (1988) Poisson approximations in selected metrics by coupling and semigroup methods. J. Stat. Plann. Inf. 20, 1–22.Google Scholar
[6] Deheuvels, P., Pfeifer, D. and Puri, M. L. (1989) A new semigroup technique in Poisson approximation. Semigroup Forum 38, 189–201.CrossRefGoogle Scholar
[7] Fishburn, P. C. (1980) Stochastic dominance and moments of distributions. Math. Operat. Res. 5, 94–100.Google Scholar
[8] Gerber, H. (1981) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph.Google Scholar
[9] Gerber, H. (1984) Error bounds for the compound Poisson approximation. Insurance: Math. Econ. 3, 191–194.Google Scholar
[10] Goovaerts, M. J., De Vylder, F. and Haezendonck, J. (1984) Insurance Premiums. North-Holland, Amsterdam.Google Scholar
[11] Hipp, C. (1985) Approximation of aggregate claims distributions by compound Poisson distributions. Insurance: Math. Econ. 4, 227–232, Correction note: 6 (1987), 165.Google Scholar
[12] Hipp, C. (1987) Improved approximations for the aggregate claims distribution in the individual model. Astin Bull. 16, 89–100.CrossRefGoogle Scholar
[13] Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables (Engl. Transl.). Noordhoff, 1971.Google Scholar
[14] Ibragimov, I. A. and Presman, E. L. (1973) On the rate of approach of the distributions of sums of independent random variables to accompanying distributions. Theory Prob. Appl. 18, 713–727.CrossRefGoogle Scholar
[15] Le Cam, L. (1986) Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. Springer-Verlag, Berlin.Google Scholar
[16] Maejima, M. and Rachev, S. T. (1987) An ideal metric and the rate of convergence to a self similar process. Ann. Prob. 15, 708–727.CrossRefGoogle Scholar
[17] Petrov, V. V. (1975) Sums of Independent Random Variables. Springer-Verlag, Berlin.Google Scholar
[18] Presman, E. L. (1983) Approximation of binomial distributions by infinitely divisible ones. Theory Prob. Appl. 28, 393–403.Google Scholar
[19] Rolski, T. (1976) Order relations in the set of probability distributions and their applications in queuing theory. Dissertationes Math., Warszawa.Google Scholar
[20] Senatov, V. V. (1980) Uniform estimates of the rate of convergence in the multidimensional central limit theorem. Theory Prob. Appl. 25, 745–759.Google Scholar
[21] Serfling, R. J. (1975) A general Poisson approximation theorem. Ann. Prob. 3, 726–731.Google Scholar
[22] Shiganov, I. S. (1986) A refinement of the upper bound of the constant in the remainder term of the central limit theorem. In Stability Problems for Stochastic Models, VNIISI, Moscow, 1982, 109–115 (in Russian); English translation, J. Soviet Math. 35 (1986), No. 3.Google Scholar
[23] Whitmore, G. A. (1970) Third degree stochastic dominance. Amer. Econ. Rev. 60, 457–459.Google Scholar
[24] Zaitsev, A. Yu. (1983) On the accuracy of approximation of distributions of sums of independent random variables that are nonzero with small probability by accompanying laws. Theory Prob. Appl. 28, 657–669.CrossRefGoogle Scholar
[25] Zolotarev, V. M. (1976) Approximation of distributions of sums of independent random variables with values in infinite dimensional spaces. Theory Prob. Appl. 21, 721–737.Google Scholar
[26] Zolotarev, V. M. (1977) Ideal metrics in the problem of approximating distributions of sums of independent random variables. Theory Prob. Appl. 22, 433–439.Google Scholar
[27] Zolotarev, V. M. (1978) On pseudomoments. Theory Prob. Appl. 23, 269–278.Google Scholar
[28] Zolotarev, V. M. (1983) Probability metrics. Theory Prob. Appl. 28, 278–302.Google Scholar
[29] Zolotarev, V. M. (1987) Modern Theory of Summation of Independent Random Variables (in Russian). Nauka, Moscow.Google Scholar