Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-17T08:10:10.413Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

Part of: Distribution theory Applications

Published online by Cambridge University Press:  14 July 2016

Michael Weba
Michael Weba*
Affiliation:
University of Applied Sciences, Fulda
*
∗Postal address: Department of Applied Computer Science, University of Applied Sciences, Fulda, Marquardstr. 35, 36039 Fulda, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p1,p2,…, pn is often approximated by a Poisson distribution with parameter λ = p1 + p2 + pn. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p1,p2,…,pn.

Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

Keywords

Binomial distributionPoisson distributiontotal variation distancerisk theoryaccumulated claim distribution

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic) 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 97 - 104
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barbour, A. D., Chen, L. H. Y., and Loh, W.-L. (1992a). Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Prob. 20, 1843–1866.Google Scholar
Barbour, A. D., and Hall, P. (1984). On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992b). Poisson Approximation. Clarendon Press, Oxford.Google Scholar
Beard, R. E., Pentikäinen, T., and Pesonen, E. (1984). Risk Theory. Chapman and Hall, London.Google Scholar
Chen, L. H. Y., and Choi, K. P. (1992). Some asymptotic and large deviation results in Poisson approximation. Ann. Prob. 20, 1867–1876.CrossRefGoogle Scholar
Chow, Y. S., and Teicher, H. (1988). Probability Theory. Springer, New York.Google Scholar
Deheuvels, P., Pfeifer, D., and Puri, M. L. (1989). A new semigroup technique in Poisson approximation. Semigroup Forum 38, 189–201.Google Scholar
Ehm, W. (1991). Binomial approximation to the Poisson binomial distribution. Statist. Prob. Lett. 11, 7–16.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Hipp, C. (1985). Approximation of aggregate claim distributions by compound Poisson distributions. Insurance: Mathematics and Economics 4, 227–232.Google Scholar
Kennedy, J. E., and Quine, M. P. (1989). The total variation distance between the binomial and the Poisson distribution. Ann. Prob. 17, 396–400.Google Scholar
LeCam, L. (1960). An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 1181–1197.Google Scholar
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.Google Scholar
Serfling, R. J. (1975). A general Poisson approximation theorem. Ann. Prob. 3, 726–731.CrossRefGoogle Scholar
Shorgin, S. Y. (1977). Approximation of a generalized binomial distribution. Theory Prob. Appl. 22, 846–850.Google Scholar
Straub, E. (1988). Non-life Insurance Mathematics. Springer, New York.Google Scholar
Witte, H.-J. (1990). A unification of some approaches to Poisson approximation. J. Appl. Prob. 27, 611–621.Google Scholar