On approximation of Markov binomial distributions

TL;DR

This paper investigates approximate distributions for the sum of a stationary Markov chain with binary states, showing that negative binomial and binomial distributions serve as suitable approximations depending on the variance-to-mean ratio, with explicit error bounds.

Contribution

It introduces practical approximation methods for the Markov binomial distribution using negative binomial and binomial models, with explicit error estimates for stationary chains.

Findings

Negative binomial approximates when variance exceeds mean

Binomial approximates when variance is less than mean

Explicit error bounds are derived for both approximations

Abstract

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.