Distribution of the scan statistic for a sequence of bistate trials

Abstract

Let S_n(r) = max_r≤t≤n ∑^t_k=t-r+1 X_k be the scan statistic of window size r for a sequence of n bistate trials {X_i}_i=1ⁿ. The scan statistic S_n(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of S_n(r) are rather complex, and they can only be numerically implemented when {X_i}_i=1ⁿ is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of S_n(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when {X_i}_i=1ⁿ is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.