Scanning Points in a Poisson Process

Abstract

Let Y_t(w) denote the number of points (X’s) in the interval {t, t + w). The scan statistic \( {S_w} = \mathop {\max }\limits_{0 < t < T - w} \;\{ {Y_t}(w)\}\), denotes the largest number of points to be found in any subinterval of [0, T) of length w. Let X₍₁₎≤ X₍₂₎≤ ...,denote the ordered values of the X’s. The statistic W_k,the size of the smallest subinterval of [0, T) that contains k points, equals \( \mathop {\min }\limits_{0 \leqslant w \leqslant T} \{ w:{S_w} \geqslant k\} \; = \mathop {\min }\limits_{1 \leqslant i} \{ {X_{(i + k - 1)}} - {X_{(i)}}\}\). For the case where the N points are uniformly distributed on [0, T),the common probabilities P(S_W≥ k) = P(W_k≤ w) are denoted P(k; N, w/T). The maximum cluster S_w is called the scan statistic, and the smallest interval W_r+1 is called the r-scan statistic.