Functional Sieve Bootstrap for the Partial Sum Process with Application to Change-Point Detection

Abstract: This paper applies the functional sieve bootstrap (FSB) to estimate the distribution of the partial sum process for time series stemming from a weakly stationary functional process. Consistency of the FSB procedure under weak assumptions on the underlying functional process is established. This result allows for the application of the FSB procedure to testing for a change-point in the mean of a functional time series using the CUSUM-statistic. We show that the FSB asymptotically correctly estimates critical values of the CUSUM-based test under the null-hypothesis. Consistency of the FSB-based test under local alternatives also is proven. The finite sample performance of the procedure is studied via simulations.