Scan Statistics of Rate Function of Scores on Poisson Point Processes

Abstract

Let X1,X2, ... be independent and identically distributed (i.i.d.) random variables with positive mean. Each random variable Xi is associated with a random time ti and t1, t2, ... are distributed according to a Poisson point process on [0, 1]. We would like to detect unusual behavior in a segment of the interval [0, 1]. For each 0 < x < y < 1, we compute a score S(x, y) which is large when there is unusual behavior in the interval [x, y]. Since x, y are unknown, we consider the maximum of these scores over 0 < x < y < 1, known in the statistical literature as scan statistics. We derive formulas for approximating the tail probabilities of these scan statistics and check them through numerical computations and simulation exercises.