Functional Sieve Bootstrap for the Partial Sum Process with Application to Change-Point Detection without Dimension Reduction

Abstract

Change-points in functional time series can be detected using the CUSUM-statistic, which is a non-linear functional of the partial sum process. Various methods have been proposed to obtain critical values for this statistic. In this paper we use the functional autoregressive sieve bootstrap to imitate the behavior of the partial sum process and we show that this procedure asymptotically correct estimates critical values under the null hypothesis. We also establish the consistency of the corresponding bootstrap based test under local alternatives. The finite sample performance of the procedure is studied via simulations under the null -hypothesis and under the alternative.