On the Asymptotic Distribution of the Scan Statistic for Empirical Distributions

TL;DR

This paper studies the asymptotic behavior of various scan statistics, including the Studentized version, for empirical distributions to detect anomalous intervals, using advanced probabilistic tools.

Contribution

It provides new theoretical insights into the asymptotic distribution of scan statistics, especially the Studentized variant, for empirical data analysis.

Findings

Asymptotic distribution characterized for different scan statistic variants.

Studentized scan statistic shown to be preferable in practice.

Theoretical framework established using Kolmogorov's theorem and Poisson approximation.

Abstract

We investigate the asymptotic behavior of several variants of the scan statistic applied to empirical distributions, which can be applied to detect the presence of an anomalous interval with any length. Of particular interest is Studentized scan statistic that is preferable in practice. The main ingredients in the proof are Kolmogorov's theorem, a Poisson approximation, and recent technical results by Kabluchko et al (2014).