Scan Statistics pp 185-199

# Scanning Points in a Poisson Process

• Joseph Glaz
• Joseph Naus
• Sylvan Wallenstein
Part of the Springer Series in Statistics book series (SSS)

## 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.

## Keywords

Poisson Distribution Poisson Process Exact Result Common Probability Exponential Random Variate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Joseph Glaz
• 1
• Joseph Naus
• 2
• Sylvan Wallenstein
• 3
1. 1.Department of Statistics The College of Liberal Arts and SciencesUniversity of ConnecticutStorrsUSA
2. 2.Department of Statistics RutgersThe State University of New JerseyPiscatawayUSA
3. 3.Department of Biomathematical SciencesMount Sinai School of MedicineNew YorkUSA