On Hipp's compound Poisson approximations via concentration functions
Bero Roos
Source: Bernoulli Volume 11, Number 3
(2005), 533-557.
Abstract
This paper is devoted to a refinement of Hipp's method in the compound Poisson approximation to the distribution of the sum of independent but not necessarily identically distributed random variables. Approximations by related Kornya-Presman signed measures are also considered. By using alternative proofs, we show that several constants in the upper bounds for the Kolmogorov and the stop-loss distances can be reduced. Concentration functions play an important role in Hipp's method. Therefore, we provide an improvement of the constant in Le~Cam's bound for concentration functions of compound Poisson distributions. But we also follow Hipp's idea to estimate such concentration functions with the help of Kesten's concentration function bound for sums of independent random variables. In fact, under the assumption that the summands are identically distributed, we present a smaller constant in Kesten's bound, the proof of which is based on a slight sharpening of Le Cam's version of the Kolmogorov-Rogozin inequality.
First Page:
Show
Hide
Keywords: compound Poisson approximation; concentration functions; explicit constants; Hipp's method; individual risk model; Kolmogorov distance; Kornya-Presman signed measures; random sums; stop-loss distance; sums of independent random variables; upper bounds
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.bj/1120591188
Mathematical Reviews number (MathSciNet): MR2147774
Digital Object Identifier: doi:10.3150/bj/1120591188
Zentralblatt MATH identifier: 1076.60036
References
[1] Arak, T.V. and Zaitsev, A.Yu. (1988) Uniform Limit Theorems for Sums of Independent Random Variables. Transl. from Russian, Proc. Steklov Inst. Math. 1. Providence, RI: American Mathematical Society.
Mathematical Reviews (MathSciNet): MR974089
Zentralblatt MATH: 0659.60070
[2] Barbour, A.D. and Xia, A. (1999) Poisson perturbations. ESAIM Probab. Statist., 3, 131-150.
Mathematical Reviews (MathSciNet): MR1716120
Digital Object Identifier: doi:10.1051/ps:1999106
[3] Barbour, A.D. and Xia, A. (2000) Estimating Stein´s constants for compound Poisson approximation. Bernoulli, 6, 581-590.
Zentralblatt MATH: 0959.62014
Digital Object Identifier: doi:10.2307/3318506
[4] Barbour, A.D., Chen, L.H.Y. and Loh, W.-L. (1992a) Compound Poisson approximation for nonnegative random variables via Stein´s method. Ann. Probab., 20, 1843-1866.
Zentralblatt MATH: 0765.60015
Digital Object Identifier: doi:10.1214/aop/1176989531
[5] Barbour, A.D., Holst, L. and Janson, S. (1992b) Poisson Approximation. Oxford: Clarendon Press.
Mathematical Reviews (MathSciNet): MR1163825
[6] Bening, V.E., Korolev, V.Yu. and Shorgin, S.Ya. (1997) On approximations to generalized Poisson distributions. J. Math. Sci. (New York), 83, 360-373.
Mathematical Reviews (MathSciNet): MR1434816
Zentralblatt MATH: 0885.60017
Digital Object Identifier: doi:10.1007/BF02400920
[7] Cekanavicius, V. (1997) Approximation of the generalized Poisson binomial distribution: Asymptotic expansions. Lithuanian Math. J., 37, 1-12.
Mathematical Reviews (MathSciNet): MR1456925
[8] Cekanavicius, V. (2003) Infinitely divisible approximations for discrete nonlattice variables. Adv. Appl. Probab., 35, 982-1006.
Mathematical Reviews (MathSciNet): MR2014266
Zentralblatt MATH: 1064.60054
Digital Object Identifier: doi:10.1239/aap/1067436331
Project Euclid: euclid.aap/1067436331
[9] Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edition. New York: Wiley.
Mathematical Reviews (MathSciNet): MR270403
[10] Hengartner, W. and Theodorescu, R. (1973) Concentration Functions. New York: Academic Press.
Mathematical Reviews (MathSciNet): MR331448
[11] Hipp, C. (1985) Approximation of aggregate claims distributions by compound Poisson distributions.
Mathematical Reviews (MathSciNet): MR810720
[12] Insurance Math. Econom., 4, 227-232. Correction note: 6, 165 (1987).
Mathematical Reviews (MathSciNet): MR896419
[13] Hipp, C. (1986) Improved approximations for the aggregate claims distribution in the individual model. ASTIN Bull., 16, 89-100.
Digital Object Identifier: doi:10.2143/AST.16.2.2015001
[14] Hipp, C. and Michel, R. (1990) Risikotheorie: Stochastische Modelle und Statistische Verfahren. Karlsruhe: Verlag Versicherungswirtschaft.
[15] Kesten, H. (1969) A sharper form of the Doeblin-Lévy-Kolmogorov-Rogozin inequality for concentration functions. Math. Scand., 25, 133-144.
[16] Kolmogorov, A.N. (1956) Two uniform limit theorems for sums of independent random variables. Theory Probab. Appl., 1, 384-394.
Digital Object Identifier: doi:10.1137/1101030
[17] Kolmogorov, A. (1958) Sur les propriétés des fonctions de concentrations de M.P. Lévy. Ann. Inst. H. Poincaré, 16, 27-34.
[18] Kornya, P.S. (1983) Distribution of aggregate claims in the individual risk theory model. Trans. Soc. Actuaries, 35, 823-858.
[19] Kruopis, J. (1986) Precision of approximation of the generalized binomial distribution by convolutions of Poisson measures. Lithuanian Math. J., 26, 37-49.
Zentralblatt MATH: 0631.60019
Digital Object Identifier: doi:10.1007/BF00971345
[20] Le Cam, L. (1965) On the distribution of sums of independent random variables. In J. Neyman and L. Le Cam (eds), Bernoulli (1713) - Bayes (1763) - Laplace (1813): Proceedings of an International Research Seminar, pp. 179-202. Berlin: Springer-Verlag.
Mathematical Reviews (MathSciNet): MR199871
[21] Le Cam, L. (1986) Asymptotic Methods in Statistical Decision Theory. New York: Springer-Verlag.
Mathematical Reviews (MathSciNet): MR856411
Zentralblatt MATH: 0605.62002
[22] Müller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. Chichester: Wiley.
[23] Nagaev, S.V. and Khodzhibagyan, S.S. (1996) On an estimate of the concentration function of sums of independent random variables. Theory Probab. Appl., 41, 560-569.
Mathematical Reviews (MathSciNet): MR1450083
[24] Panjer, H.P. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bull., 12, 22-26.
Mathematical Reviews (MathSciNet): MR632572
[25] Petrov, V.V. (1975) Sums of Independent Random Variables. Berlin: Springer-Verlag.
Mathematical Reviews (MathSciNet): MR388499
[26] Petrov, V.V. (1995) Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford: Clarendon Press.
Mathematical Reviews (MathSciNet): MR1353441
[27] Presman, É.L. (1983) Approximation of binomial distributions by infinitely divisible ones. Theory Probab. Appl., 28, 393-403.
Zentralblatt MATH: 0533.60018
Digital Object Identifier: doi:10.1137/1128033
[28] Rogozin, B.A. (1961) An estimate for concentration functions. Theory Probab. Appl., 6, 94-97.
Mathematical Reviews (MathSciNet): MR131893
[29] Roos, B. (2001) Sharp constants in the Poisson approximation. Statist. Probab. Lett., 52, 155-168.
Mathematical Reviews (MathSciNet): MR1841404
[30] Roos, B. (2002) Kerstan´s method in the multivariate Poisson approximation: an expansion in the exponent. Theory Probab. Appl., 47, 358-363.
Zentralblatt MATH: 1033.60029
Digital Object Identifier: doi:10.1137/S0040585X97979779
[31] Roos, B. (2003) Kerstan´s method for compound Poisson approximation. Ann. Probab., 31, 1754- 1771.
Zentralblatt MATH: 1041.62011
Digital Object Identifier: doi:10.1214/aop/1068646365
[32] Salikhov, N.P. (1996) An estimate of the concentration function by the Esseen method. Theory Probab. Appl., 41, 504-518.
Mathematical Reviews (MathSciNet): MR1450074
[33] Zaitsev, A.Yu. (1983) On the accuracy of approximation of distributions of sums of independent random variables - which are non-zero with a small probability - by means of accompanying laws. Theory Probab. Appl., 28, 657-669.
Mathematical Reviews (MathSciNet): MR726889
2013 © Bernoulli Society for Mathematical Statistics and Probability
