Monitoring panels of sparse functional data

Panels of random functions are common in applications of functional data analysis. They often occur when sequences of functions are observed at a number of different locations. We propose a methodology to monitor for structural breaks in such panels and to identify the changing components with statistical certainty. Our approach relies on a Full-CUSUM statistic that has proved to be powerful in finite dimensions but has not been applied to functional data. To account for the practically relevant problem of sparsity, we formulate our results for triangular arrays of nonstationary, sparse estimators. The derivation of our asymptotic theory relies on new Gaussian approximations on the Banach space of continuous functions, which imply new convergence results for the change point detectors. We illustrate our approach with a simulation study and application to intraday returns on exchange traded funds.