Corrected Discrete Approximations for the Conditional and Unconditional Distributions of the Continuous Scan Statistic

TL;DR

This paper develops corrected discrete approximation methods for the distribution of the continuous scan statistic in Poisson processes, improving accuracy through a change-of-measure approach and Richardson's extrapolation, with numerical validation.

Contribution

It introduces a first-order discrete approximation and a second-order corrected version for the scan statistic distribution, enhancing precision over existing methods.

Findings

Corrected approximation outperforms uncorrected methods in numerical tests.

First-order approximation involves functionals of the Poisson process.

Richardson's extrapolation effectively improves approximation accuracy.

Abstract

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). With the help of a change-of-measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.