Abstract: We study sequential monitoring procedures that detect instabilities of the regression operator in an underlying (fully) functional regression model allowing for dependence. These open-end and closed-end procedures are built on a functional principal components analysis of both the predictor and response functions, thus giving rise to multivariate detector functions, whose fluctuations are compared against a curved threshold function. The main theoretical result of the paper quantifies the large-sample behavior of the procedures under the null hypothesis of a stable regression operator. To establish these limit results, classical results on functional principal components analysis are generalized to a dependent setting, which may be of interest in its own sake. In an accompanying empirical study we illustrate the finite sample properties, while an application to environmental data highlights practical usefulness. To the best of our knowledge this is the first paper that combines sequential with functional data methodology.

Key words and phrases: Change-point analysis, dependent functional data, functional data analysis, functional principal components, sequential methodology, stopping times, structural breaks.