Practical properties of the CUSUM process

TL;DR

This paper investigates the properties of the CUSUM process, providing new theoretical insights, computational methods, and applications to change-point detection and queuing probabilities.

Contribution

It introduces new properties, recursive algorithms, and bounds for the CUSUM process, enhancing its application in change detection and queuing analysis.

Findings

Derived explicit expressions for CUSUM's moment generating function and moments.

Developed fast recursive algorithms for CUSUM computations.

Provided bounds and asymptotes for CUSUM related probabilities.

Abstract

We explore the behavior and establish new properties of the cumulative-sum process (CUSUM) and its running maximum. The study includes precise expressions for CUSUM's moment generating function and moments, fast recursive computing algorithms, lower and upper bounds, as well as asymptotes. Results are applied to single, multiple, and transient change-point problems, for the calculation of thresholds that provide a desired control of familywise false alarm rates, as well as the quantiles of queuing processes and probabilities of their large deviation at least once over a given time interval.