Adaptive Estimation for Weakly Dependent Functional Times Series

ABSTRACT We propose adaptive mean and autocovariance function estimators for stationary functional time series under ‐approximability assumptions. These estimators are designed to adapt to the regularity of the curves and to accommodate both sparse and dense data designs. The sample paths are observed with error at possibly random design points. Data‐driven local bandwidths are selected by minimizing explicit quadratic risk bounds that exploit the local regularity of the process. As a first step, we introduce a local regularity estimator and derive a nonasymptotic concentration bound for it. We also derive the asymptotic normality of the mean estimator, which allows honest inference for irregular mean functions. Simulations and a real data application illustrate the performance of the new estimators.