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On Negative Binomial Approximation to k-Runs

Part of: Combinatorial probability Distribution theory - Probability Limit theorems

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
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Abstract

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

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