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This article considers testing for mean-level shifts in functional data. Theclass of the famous Darling-Erd\H{o}s-type cumulative sums (CUSUM) proceduresis extended to functional time series under short range dependence conditionswhich are satisfied by functional analogues of many popular time series modelsincluding the linear functional AR and the non-linear functional ARCH. Wefollow a data driven, projection-based approach where the lower-dimensionalsubspace is determined by (long run) functional principal components which areeigenfunctions of the long run covariance operator. This second-order structureis generally unknown and estimation is crucial - it plays an even moreimportant role than in the classical univariate setup because it generates thefinite-dimensional subspaces. We discuss suitable estimates and demonstrateempirically that altogether this change-point procedure performs well undermoderate temporal dependence.  Moreover, Darling-Erd\H{o}s-type change-point estimates based on (long run)functional principal components as well as the corresponding "fully-functional"counterparts are provided and the testing procedure is finally applied topublicly accessible electricity data from a German power company.

A Darling-Erd\H{o}s-type CUSUM-procedure for functional data II