Improvements in the Poisson approximation of mixed Poisson distributions

Journal of Statistical Planning and Inference

Volumn 113, Issue 2, 2003, Pages 467-483

  Roos, Bero ^a  

^a UNIVERSITY OF HAMBURG   (Germany)

Author keywords

Asymptotic relations; Finite signed measures; Mixed Poisson distribution; Negative binomial distribution; Poisson approximation; Probability metrics; Sharp constant; Total variation distance; Weak moment conditions

Indexed keywords

EID: 0037408743     PISSN: 03783758     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-3758(02)00095-2     Document Type: Article

References (27)

1. Barbour, A.D., 1987. Asymptotic expansions in the Poisson limit theorem. Ann. Probab. 15, 748-766.
2. Barbour, A.D., Holst, L., Janson, S., 1992. Poisson Approximation. Clarendon Press, Oxford.
3. Chihara, T.S., 1978. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
4. Daley, D.J., Vere-Jones, D., 1988. An Introduction to the Theory of Point Processes. Springer, New York.
5. Deheuvels, P., Pfeifer, D., 1986a. A semigroup approach to Poisson approximation. Ann. Probab. 14, 663-676.
6. Deheuvels, P., Pfeifer, D., 1986b. Operator semigroups and Poisson convergence in selected metrics. Semigroup Forum 34, 203-224. Errata: 35, 251, 1987.
7. Deheuvels, P., Pfeifer, D., 1988. On a relationship between Uspensky's theorem and Poisson approximation. Ann. Inst. Statist. Math. 40, 671-681.
8. Douglas, J.B., 1980. Analysis with Standard Contagious Distributions. International Co-operative Publishing House, Fairland, MD.
9. Feller, W., 1943. On a general class of contagious distributions. Ann. Math. Statist. 14, 389-400.
10. Freedman, D., 1974. The Poisson approximation for dependent events. Ann. Probab. 2, 256-269.
11. Gerber, H.U., 1984. Error bounds for the compound Poisson approximation. Insurance Math. Econom. 3, 191-194.
12. Grandell, J., 1997. Mixed Poisson Processes. Chapman & Hall, London.
13. Haight, F.A., 1967. Handbook of the Poisson Distribution. Wiley, New York.
14. Johnson, N.L., Kotz, S., Kemp, A.W., 1993. Univariate Discrete Distributions, 2nd Edition. Wiley, New York.
15. Matsunawa, T., 1982. Some strong epsilon-equivalence of random variables. Ann. Inst. Stat. Math. 34, 209-224.
16. Mitrinović, D.S., 1970. Analytic Inequalities. Springer, Berlin.
17. Pfeifer, D., 1985. Approximation-theoretic aspects of probabilistic representations for operator semigroups. J. Approx. Theory 43, 271-296.
18. Pfeifer, D., 1987. On the distance between mixed Poisson and Poisson distributions. Statist. Decisions 5, 367-379.
19. Rachev, S.T., 1991. Probability Metrics and Stability of Stochastic Models. Wiley, Chichester.
20. Romanowska, M., 1977. A note on the upper bound for the distance in total variation between the binomial and the Poisson distribution. Statist. Neerlandica. 31, 127-130.
21. Roos, B., 1999. Asymptotics and sharp bounds in the Poisson approximation to the Poisson binomial distribution. Bernoulli 5, 1021-1034.
22. Roos, B., 2000. Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion Theory Probab. Appl. 45, 258-272.
23. Roos, B., 2001. Sharp constants in the Poisson approximation. Statist. Probab. Lett. 52, 155-168.
24. Shorgin, S.Y., 1977. Approximation of a generalized binomial distribution. Theory Probab. Appl. 22, 846-850.
25. Vervaat, W., 1969. Upper bounds for the distance in total variation between the binomial or negative binomial and the Poisson distribution. Statist. Neerlandica 23, 79-86.
26. Whittaker, E.T., Watson, G.N., 1927. A Course of Modern Analysis, 4th Edition. Cambridge University Press, London.
27. Yannaros, N., 1991. Poisson approximation for random sums of Bernoulli random variables. Statist. Probab. Lett. 11, 161-165.