Scan Statistics for Integer-Valued Random Variables: Conditional Case

In this chapter, we review approximations and inequalities for the distribution of conditional scan statistics for a sequence of independent and identically distributed nonnegative integer-valued random variables, modeled by a one-parameter natural exponential family of distributions, when the total sum of the random variables is known. The distribution of conditional scan statistics is based on the joint distribution of moving sums of components of a random vector from a related multivariate discrete distribution. In most cases the exact distribution of conditional scan statistics is unknown. Therefore, accurate approximations and inequalities for their distributions are of great importance. In this chapter, we present accurate product-type and Poisson-type approximations and Bonferroni-type inequalities for the tail probabilities and expected size of conditional scan statistics for the binomial, Poisson, and negative binomial models. We also discuss the extension of the conditional scan statistics to a conditional multiple occurrence scan statistic. Numerical results are presented to evaluate the accuracy of the approximations and inequalities discussed in this chapter.