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.