Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T04:20:27.739Z Has data issue: false hasContentIssue false
  • English
  • Français

On Negative Binomial Approximation to k-Runs

Part of: Limit theorems Distribution theory - Probability Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Xiaoxin Wang  and
Aihua Xia
Xiaoxin Wang*
Affiliation:
The University of Melbourne
Aihua Xia*
Affiliation:
The University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia. Email address: danawang_79@hotmail.com
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia. Email address: danawang_79@hotmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suitable approximate model with reasonable approximation accuracy. In this paper we demonstrate that the negative binomial distribution is the most suitable approximate model for the number of k-runs: it outperforms the Poisson approximation, the general compound Poisson approximation as observed in Eichelsbacher and Roos (1999), and the translated Poisson approximation in Rollin (2005). In particular, its accuracy of approximation in terms of the total variation distance improves when the number of experiments increases, in the same way as the normal approximation improves in the Berry-Esseen theorem.

Keywords

Poisson approximationtranslated Poisson approximationcompound Poisson approximationnegative binomial approximationStein's methodtotal variation distance

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability 60E15: Inequalities; stochastic orderings

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 456 - 471
Copyright
Copyright © Applied Probability Trust 2008 

References

Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403–434.Google Scholar
Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Barbour, A. D. and Xia, A. (1999). Poisson Perturbations. ESAIM Prob. Statist. 3, 131–150.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Brown, T. C. and Xia, A. (2001). Stein's method and birth–death processes. Ann. Prob. 29, 1373–1403.CrossRefGoogle Scholar
Chiang, D. T. and Niu, S. C. (1981). Reliability of consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 30, 87–89.Google Scholar
Chryssaphinou, O. and Papastavridis, S. (1990). Limit distributions for a consecutive-k-out-of-n: F system. Adv. Appl. Prob. 22, 491–493.Google Scholar
Eichelsbacher, P. and Roos, M. (1999). Compound Poisson approximation for dissociated random variables via Stein's method. Combin. Prob. Comput. 8, 335–346.Google Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions. John Wiley, New York.CrossRefGoogle Scholar
Koutras, M. V. and Alexandrou, V. A. (1997). Non-parametric randomness tests based on success runs of fixed length. Statist. Prob. Lett. 32, 393–404.CrossRefGoogle Scholar
Koutras, M. V. and Papastavridis, S. G. (1993). Application of the Stein–Chen method for bounds and limit theorems in the reliability of coherent structures. Naval Res. Logistics 40, 617–631.Google Scholar
Mood, A. M. (1940). The distribution theory of runs. Ann. Math. Statist. 11, 367–392.Google Scholar
Papastavridis, S. (1987). A limit theorem for the reliability of a consecutive-k-out-of-n system. Adv. Appl. Prob. 19, 746–748.CrossRefGoogle Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford.Google Scholar
Rollin, A. (2005). Approximation of sum of conditionally independent variables by the translated Poisson distribution. Bernoulli 11, 1115–1128.Google Scholar
Roos, M. (1994). Stein–Chen method for compound Poisson approximation. Doctoral Thesis, University of Zürich.Google Scholar
Von Mises, R. (1921). Das Problem der Iterationen. Z. Angewandte Math. Mech. 1, 298–307.Google Scholar