Title: Testing Equality of Autocovariance Operators for Functional Time Series,
Abstract: We consider strictly stationary stochastic processes of Hilbert space-valued
random variables and focus on tests of the equality of the lag-zero
autocovariance operators of several independent functional time series. A
moving block bootstrap-based testing procedure is proposed which generates
pseudo random elements that satisfy the null hypothesis of interest. It is
based on directly bootstrapping the time series of tensor products which
overcomes some common difficulties associated with applications of the
bootstrap to related testing problems. The suggested methodology can be
potentially applied to a broad range of test statistics of the hypotheses of
interest. As an example, we establish validity for approximating the
distribution under the null of a fully functional test statistic based on the
Hilbert-Schmidt distance of the corresponding sample lag-zero autocovariance
operators, and show consistency under the alternative. As a prerequisite, we
prove a central limit theorem for the moving block bootstrap procedure applied
to the sample autocovariance operator which is of interest on its own. The
finite sample size and power performance of the suggested moving block
bootstrap-based testing procedure is illustrated through simulations and an
application to a real-life dataset is discussed.