Statistica Sinica 23 (2013), 1541-1551

THE MAXIMUM OF A RATCHET SCANNING PROCESS

OVER A POISSON RANDOM FIELD

I-Ping Tu

Academia Sinica

Abstract: Scan statistic is a popular method in searching non-random clusters over some random field. Motivated by a high energy particle detection problem, we are interested in a ratchet scan statistic whose scanning window is a grid set, instead of a rectangular box. We further generalize it to an $r$-dimensional problem for any $r\in\mathbb{Z}^+$ and provide a tail probability approximation for a ratchet scan statistic over an $r$-dimensional homogeneous Poisson process. We show that the ratchet effect can be factored out as an overshoot function $\nu$ in each dimension.

Key words and phrases: Cosmic ray, high dimension, Poisson process, ratchet, scan statistic, tail probability approximation.