On approximation of Markov binomial distributions
Aihua Xia and Mei Zhang
Source: Bernoulli Volume 15, Number 4
(2009), 1335-1350.
Abstract
For a Markov chain X={Xi, i=1, 2, …, n} with the state space {0, 1}, the random variable S:=∑i=1nXi is said to follow a Markov binomial distribution. The exact distribution of S, denoted , is very computationally intensive for large n (see Gabriel [Biometrika 46 (1959) 454–460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677–680]) and this paper concerns suitable approximate distributions for
when X is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for
when Var S is greater than and less than
, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.
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Keywords: binomial distribution; coupling; Markov binomial distribution; negative binomial distribution; Stein’s method; total variation distance
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.bj/1262962238
Digital Object Identifier: doi:10.3150/09-BEJ194
Mathematical Reviews number (MathSciNet): MR2597595
Zentralblatt MATH identifier: 05816144
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