On approximation of Markov binomial distributions

Details On approximation of Markov binomial distributions On approximation of Markov binomial distributions

2010-01-13

arxiv.org/abs/1001.2144v1

Bernoulli 2009, Vol. 15, No. 4, 1335-1350

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/09-BEJ194 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/09-BEJ194

For a Markov chain $\mathbf{X}=\{X_i,i=1,2,...,n\}$ with the state space $\{0,1\}$, the random variable $S:=\sum_{i=1}^nX_i$ is said to follow a Markov binomial distribution. The exact distribution of $S$, denoted $\mathcal{L}S$, 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 $\mathcal{L}S$ when $\mathbf{X}$ is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for $\mathcal{L}S$ when $\operatorname {Var}S$ is greater than and less than $\mathbb{E}S$, 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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