A Nonparametric Regression Spectrum : Estimation, Asymptotic Properties and Data Analysis

Classical spectral analysis in statistics considers decomposition of stationary time series into sinusoidal components. The autocovariance and the spectrum are fundamental elements for analyzing a given time series both in time and frequency domain. However, in practice one frequently observes nonstationary time series. In order to apply spectral analysis to these processes, an extension of the classical spectral theory to more general situations is required. This thesis investigates dependence structures in multivariate time series that are characterized by deterministic trends. Here, we extend the theory of stationary processes to deterministic nonparametric trend functions. In a nonparametric regression setting these functions are usually unknown and have to be estimated. Estimation of the trend function will be performed by applying wavelet thresholding, a simple but yet efficient way to recover a signal of unknown regularity from some noisy data. Chapter 2 presents a review about wavelets and their use in statistics. This involves construction of compactly supported wavelet bases, wavelet transformation of a square integrable function and the application in linear and nonlinear function estimation. An extensive review of the literature on wavelet thresholding is presented and some asymptotic results are derived. In chapter 3, we consider dependence structures in multivariate time series that are due to similarities in underlying deterministic trends. Results from spectral analysis for stationary processes are extended to deterministic trend