Test for the mean of high-dimensional functional time series

The one-sample test and two-sample test for the mean of high-dimensional functional time series are considered in this study. The proposed tests are built on the dimension-wise max- norm of the sum of squares of diverging projections. The null distribution of the test statistics is investigated using normal approximation, and the asymptotic behavior under the alternative is studied. The approach is robust to the cross-series dependence of unknown forms and magnitude. To approximate the critical values, a blockwise wild bootstrap method for functional time series is employed. Both fully and partially observed data are analyzed in theoretical research and numerical studies. Evidence from simulation studies and an IT stock data case study demonstrates the usefulness of the test in practice. The proposed methods have been implemented in a R package.