PhD Candidate, Statistics
Complex time series with features, such as non-linearity, high-dimensionality and functional structures, have inspired many interests in statistics community due to limitations of traditional time series models and advancement of methodology and theory of nonparametric statistics. In this dissertation, the nonparametric models for such complex time series are studied. For modeling the financial volatility, we proposed estimators for semiparametric GARCH models with additive autoregressive components linked together by a dynamic coefficient based on spline smoothing. The proposed estimators are computationally efficient and theoretically reliable. The performance of our method is evaluated by various simulated processes and a real financial return series. For modeling and forecasting the functional time series, we combined the functional principal component (FPC) analysis with time series modeling to achieve the smoothing, dimension reduction and prediction with expedient computation. Extensive simulation studies have been conducted to compare the prediction accuracy of our method with other competing methods. Application of the proposed procedure to the yield curves of US Treasury bond has produced superior out-of-sample forecasts. For comparing the functional derivatives of regression functions from two groups, we developed a novel method to construct simultaneous confidence bands for the difference of derivatives. This method was derived to answer a question arising from climate: Is the temperature transition in Athens, Georgia different under the substantial global atmospheric pressure oscillations? We show that the proposed spline confidence bands are asymptotically efficient. The performance of the confidence bands is illustrated through numerical simulation studies and a temperature data collected in Athens, GA, in US.