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SOME NEW BOUNDS AND APPROXIMATIONS ON TAIL PROBABILITIES OF THE POISSON AND OTHER DISCRETE DISTRIBUTIONS

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  11 October 2018

Steven G. From  and
Andrew W. Swift
Steven G. From
Affiliation:
Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-0243, USA E-mail: [email protected]; [email protected]
Andrew W. Swift
Affiliation:
Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-0243, USA E-mail: [email protected]; [email protected]
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Abstract

In this paper, we discuss new bounds and approximations for tail probabilities of certain discrete distributions. Several different methods are used to obtain bounds and/or approximations. Excellent upper and lower bounds are obtained for the Poisson distribution. Excellent approximations (and not bounds necessarily) are also obtained for other discrete distributions. Numerical comparisons made to previously proposed methods demonstrate that the new bounds and/or approximations compare very favorably. Some conjectures are made.

Keywords

approximations and boundsdiscrete distributionsPoisson distributiontail probabilities

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E15: Exact distribution theory
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 1 , January 2020 , pp. 53 - 71
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Bohman, H. (1963) Two inequalities for Poisson distributions. Skandinavisk Aktuarietidskrift 46: 47–52.Google Scholar
2.Consul, P.C. (1989) Generalized Poisson distributions volume 99 of Statistics: Textbooks and Monographs. New York: Marcel Dekker, Inc., Properties and applications.Google Scholar
3.Dyer, J.S. & Owen, A.B. (2012) Correct ordering in the Zipf-Poisson ensemble. Journal of the American Statistical Association 107(500): 1510–1517.Google Scholar
4.Friendly, M (2017) vcdExtra: ‘vcd’ Extensions and Additions. R package version 0.7-1.Google Scholar
5.From, S.G. (2016) Some new generalizations of Jensen's inequality with related results and applications. The Australian Journal of Mathematical Analysis and Applications 13(1): 29, Art. 1.Google Scholar
6.From, S.G. (2017) Some new inequalities of Hermite-Hadamard and Fejér type for certain functions with higher convexity. The Australian Journal of Mathematical Analysis and Applications 14(1): 17, Art. 10.Google Scholar
7.Gross, A.J. & Hosmer, D.W. Jr., (1978) Approximating tail areas of probability distributions. The Annals of Statistics 6(6): 1352–1359.Google Scholar
8.Johnson, N.L., Kotz, S., & Kemp, A.W. (1992) Univariate discrete distributions. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, 2nd ed. New York: John Wiley & Sons, Inc., A Wiley-Interscience Publication.Google Scholar
9.Klar, B (1999) Goodness-of-fit tests for discrete models based on the integrated distribution function. Metrika. International Journal for Theoretical and Applied Statistics 49(1): 53–69.Google Scholar
10.Klar, B (2000) Bounds on tail probabilities of discrete distributions. Probability in the Engineering and Informational Sciences 14(2): 161–171.Google Scholar
11.Moraes, A., Tempone, R., & Vilanova, P. (2014) Hybrid Chernoff tau-leap. Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal 12(2): 581–615.Google Scholar
12.R Core Team (2017) R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.Google Scholar
13.Ross, N (2013) Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Advances in Applied Probability 45(3): 876–893.Google Scholar
14.Ross, S.M. (1998) Using the importance sampling identity to bound tail probabilities. Probability in the Engineering and Informational Sciences 12(4): 445–452.Google Scholar
15.Statisticat & LLC (2017) LaplacesDemon: Complete Environment for Bayesian Inference. R package version 16.1.0.Google Scholar