| 讲座简介： | Inference tools are developed for the time-varying mean function of functional time series. An “infeasible” B-spline estimator is proposed when all random trajectories are completely recorded without any error. Different from existing works on simultaneous inference for functional time series where confidence bands are built on classical central limit theory, we establish a Gaussian approximation for the standardized maximal deviation of the proposed “infeasible” estimator, rather than deriving its limit distribution via weak convergence in Banach space, leading to an “infeasible” simultaneous confidence region (SCR) of the time-varying bivariate mean function. When observations are only on discrete points with measurement errors, a two-step data-driven estimator is also proposed, equivalent to a tensor-product bivariate spline estimator. Under mild conditions, the two-step estimator is oracally efficient in the sense that it is asymptotically equivalent to the infeasible estimator. Asymptotically correct SCR with adaptive width for the bivariate mean function is constructed, the size of which is uniformly a factor of (logT)^(1/2) wider than pointwise confidence intervals, aided by a meticulous analysis of the extremes of the approximating Gaussian processes. Various extensions are also studied, including simultaneous inference for marginal univariate mean functions, and additivity test for the bivariate mean. Extensive simulation results strongly support the theoretical results, and a fertility rate example and temperature curve example illustrate the use of our methods. |
