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

This article considers testing for mean-level shifts in functional data. The\nclass of the famous Darling-Erd\\H{o}s-type cumulative sums (CUSUM) procedures\nis extended to functional time series under short range dependence conditions\nwhich are satisfied by functional analogues of many popular time series models\nincluding the linear functional AR and the non-linear functional ARCH. We\nfollow a data driven, projection-based approach where the lower-dimensional\nsubspace is determined by (long run) functional principal components which are\neigenfunctions of the long run covariance operator. This second-order structure\nis generally unknown and estimation is crucial - it plays an even more\nimportant role than in the classical univariate setup because it generates the\nfinite-dimensional subspaces. We discuss suitable estimates and demonstrate\nempirically that altogether this change-point procedure performs well under\nmoderate temporal dependence.\n Moreover, Darling-Erd\\H{o}s-type change-point estimates based on (long run)\nfunctional principal components as well as the corresponding "fully-functional"\ncounterparts are provided and the testing procedure is finally applied to\npublicly accessible electricity data from a German power company.\n

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