DISCRETE SCAN STATISTICS GENERATED BY EXCHANGEABLE BINARY TRIALS

摘要

Let {X_i}_i~n=1 be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable S_(n,m), to be the maximum number of Is within any m consecutive trials in {X_i}_i~n=1. The random variable S_(n,m) is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {X_i}_(i-1)~n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of S_(n,m) for 2m > n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.