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Motivated by two examples concerning global warming and monthly total import and export by China, we study time series that contain a nonparametric periodic component with an unknown period, a nonparametric trending behavior and also additive covariate effects. Further, as the amplitude function may change at some known or unknown change-point(s), we extend our model to take this dynamical periodicity into account and introduce two change-point estimators. To the best of knowledge, this is the first work to study such complex periodic structure. A two-step estimation procedure is proposed to estimate accurately the periodicity, trend and covariate effects. First, we estimate the period with the trend and covariate effects being approximated by B-splines rather than being ignored. To achieve robustness we employ a penalized M-estimation method which uses post model selection inference ideas. Next, given the period estimate, we estimate the amplitude, trend and covariate effects. Asymptotic properties of our estimators are derived, including consistency of the period estimator and asymptotic normality and oracle property of the estimated periodic sequence, trend and covariate effects. Simulation studies confirm superiority of our method and illustrate good performance of our change-point estimators. Applications to the two motivating examples demonstrate utilities of our methods.

Robust Inference for Nonstationary Time Series with Possibly Multiple Changing Periodic Structures