Practical properties of the CUSUM process

TL;DR

This work develops a comprehensive, computable theory for the CUSUM process and its running maximum by extending Spitzer's identity to obtain exact moments and MGFs, and by exploiting Bell polynomials for fast recursive and matrix computations. It reveals a linear asymptote in the MGF at a critical root $\lambda^{*}>0$ of $\mathbb{E}e^{\lambda Y}=1$ and classifies three asymptotic regimes for MGFs, enabling precise upper and lower bounds via maximal inequalities and large deviations. The results yield universal, distribution-free thresholds for abrupt and transient change-point detection, provide tighter bounds using a discrepancy measure $D_{F,G}$, and extend to randomly stopped sequences and queuing systems, with practical algorithms for computing thresholds. Collectively, the paper delivers rigorous, implementable tools for sequential detection, stopping-time comparison, and applications in queueing theory, underpinned by exact moments, MGFs, and efficient computation.

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.

Figures (5)

• Figure 1: CUSUM's exact exponential moments for various detectable differences $\delta$.
• Figure 2: CUSUM's mean and variance for various detectable differences $\delta$.
• Figure 3: CUSUM's moment generating functions for $\lambda < \lambda^{\!\ast}$ (left) and for $\lambda > \lambda^{\!\ast}$ (right).
• Figure 4: Thresholds for the transient change-point detection and their proposed upper and lower bounds. Eight scenarios correspond to eight detectable differences $\delta$ from 0.1 to 4.0. Thresholds are calculated to control the false alarm probability within $\alpha=0.05$, and the meaning of upper and lower bounds is explained in Section 6.6.
• Figure 5: Threshold $h$ as a function of the detectable difference $\delta$, for different sample sizes $n$

Theorems & Definitions (24)

• Theorem 2.1: Exact expressions for the moments and MGF
• proof
• Theorem 3.1: Recursive computation of CUSUM's moments and MGF
• proof
• Theorem 4.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 4.2