Moving block and tapered block bootstrap for functional time series with an application to the $K$-sample mean problem

摘要

We consider infinite-dimensional Hilbert space-valued random variables thatare assumed to be temporal dependent in a broad sense. We prove a central limittheorem for the moving block and the tapered block bootstrap and show thatthese block bootstrap procedures also provide consistent estimators of thespectral density operator of the underlying functional process at frequencyzero. Furthermore, we consider block bootstrap-based procedures for fullyfunctional testing of the equality of mean functions between severalindependent functional time series. We establish the validity of the blockbootstrap methods in approximating the distribution of the statistic ofinterest under the null. The finite sample behaviour of the procedures isinvestigated by means of simulations. An application to a real-life dataset isalso discussed.