Statistical inference for mean function of partially observed functional time series

We develop a statistical framework of inference for mean functions of partially observed functional time series. In the ideal case where curves are fully recorded during the observation period without noise, we establish the weak convergence in the Skorokhod space for an "ideal" estimator that may exhibit discontinuities. In the practical setting where data are contaminated by measurement errors, we propose a B-spline estimator and obtain the asymptotic distribution of its maximum deviation via Gaussian approximation techniques. Building on the established asymptotic results, we develop various forms of statistical inference under the supremum norm, including simultaneous confidence bands, two-sample tests, and tests of relevant hypotheses. For implementation, we employ the multiplier bootstrap to approximate the limiting distribution and establish the consistency of the bootstrap procedures. Numerical experiments confirm the validity of the theoretical results. The proposed methods are applied to an electroencephalogram dataset collected from visual stimulus experiments, discovering multiple scientific facts.