CUSUMtype

CUSUMtype refers to a family of statistical methods based on cumulative sums of observed deviations and is used for sequential analysis and change-point detection. These methods accumulate evidence over time to detect small, persistent changes in a process parameter, typically the mean, but they can be adapted to other moments or to nonparametric settings.

A typical CUSUMtype statistic updates recursively as new data arrive. For a sequence of observations X1, X2, ..., with a reference value μ0 and a positive decision lag k, the one-sided statistic may be defined as Sn = max(0, Sn-1 + (Xn - μ0 - k)), with S0 = 0. A signal is triggered when Sn exceeds a pre-set threshold h. Variants include one-sided and two-sided charts, and formulations based on residuals, likelihood ratios, or more general score statistics. CUSUMtype methods are closely related to cumulative sums of deviations and are designed to be highly sensitive to small shifts.

History and related approaches: CUSUMtype charts originated in quality control, notably in Page’s CUSUM chart introduced in 1954. Related methods include generalized likelihood ratio CUSUMs and variants that incorporate exponential weighting. The closely related exponentially weighted moving average (EWMA) chart shares the goal of rapid detection with smooth memory of past data.

Applications and considerations: CUSUMtype procedures are widely used for industrial process monitoring, online anomaly detection in IT systems, finance for regime shift detection, and environmental or biomedical monitoring. Key considerations include choosing a reference μ0, sensitivity parameter k, and threshold h to balance detection speed against false alarms, as well as robustness to autocorrelation and nonstationarity.