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On approximation of Markov binomial distributions

Abstract

For a Markov chain X={Xi,i=1,2,...,n}\mathbf{X}=\{X_i,i=1,2,...,n\} with the state space {0,1}\{0,1\}, the random variable S:=βˆ‘i=1nXiS:=\sum_{i=1}^nX_i is said to follow a Markov binomial distribution. The exact distribution of SS, denoted LS\mathcal{L}S, is very computationally intensive for large nn (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 LS\mathcal{L}S when X\mathbf{X} is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for LS\mathcal{L}S when Var⁑S\operatorname {Var}S is greater than and less than ES\mathbb{E}S, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ194 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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