# Approximations of the distributions of scan statistics of Poisson
processes

#### by

## Sven Erick Alm

#### Uppsala University, Dept. of Mathematics

##### In Scan Statistics and Applications, Eds Glaz&Balakrishnan, Birkhäuser (1999)

### Abstract

We study scan statistics of Poisson processes in one and higher dimensions.
First, a very accurate approximation is established in one dimension. This
is done by studying upcrossings of the scanning process and noting that these
occur in clusters. The clusters appear more or less independently and the
cluster size is estimated by a random walk argument.
This idea is then used repeatedly to obtain approximations in higher
dimensions. Simulation is used to check the accuracy of the approximations in
two and three dimensions. A discussion of these simulations is included, as
they are by no means trivial to perform.

#### Key words:

Scan statistic, Poisson process, higher dimensions, unconditional, simulation.

1999-09-16, Sven Erick Alm, sea@math.uu.se