A New Approach to Statistical Inference for Functional Time Series

The analysis of time‐indexed functional data plays an important role in the field of business and economic statistics. In the literature, statistical inference for functional time series often involves reducing the dimension of functional data to a finite dimension K$$ K $$, followed by the use of tools from multivariate analysis. The effectiveness of such an approach hinges on certain assumptions that are difficult to check in practice, and also, the results can be sensitive to the choice of K$$ K $$. In this article, we propose a fully functional approach based on sample splitting and illustrate it for several testing problems, including one and two‐sample mean testing and change point testing. Asymptotic properties of the new test statistics are derived under both the null and local alternatives in the general setting of Hilbert space‐valued time series. Simulation studies and a real data example are also presented to demonstrate the encouraging finite sample performance of the proposed tests.