Monitoring time series with short detection delay

In this paper, we develop sequential tests for a change in the mean of a dependent, Banach space-valued time series. For this purpose, we introduce a new class of weighted CUSUM statistics tailored to the detection of changes shortly after they occur. Unlike current alternatives – which either experience long detection delays or offer short delays only at the very beginning of the monitoring period – our approach provides consistently short detection delays anywhere in the monitoring period. This property is highly relevant for modern applications, such as epidemiology and finance, where short delays are crucial and the timing of the change is unpredictable. Our theoretical results are based on new Hölderian invariance principles that we prove under some high level conditions for Banach space-valued data. We show that these conditions hold in many instances and discuss as one particular example m-approximable time series on Hilbert spaces. Such time series cover important classes of models for multivariate and functional data. The resulting Hölderian invariance principle for m-approximable time series is of independent interest. A simulation study and several data examples underline the superior performance of our approach compared to existing alternatives.