A Nonparametric Regression Spectrum : Estimation, Asymptotic Properties and Data Analysis
A Nonparametric Regression Spectrum : Estimation, Asymptotic Properties and Data Analysis
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非参数回归谱：估计、渐近性质和数据分析
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发表时间：
2007
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通讯作者：
M. Heiler
中科院分区：
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作者：
M. Heiler
统计学中的经典谱分析考虑将平稳时间序列分解为正弦分量。自协方差和频谱是在时域和频域中分析给定时间序列的基本元素。然而，在实践中，人们经常观察到非平稳时间序列。为了将谱分析应用于这些过程，需要将经典谱理论扩展到更一般的情况。本论文研究了以确定性趋势为特征的多元时间序列中的依赖结构。在这里，我们将平稳过程理论扩展到确定性非参数趋势函数。在非参数回归设置中，这些函数通常是未知的并且必须进行估计。趋势函数的估计将通过应用小波阈值来执行，这是一种从一些噪声数据中恢复未知规律性信号的简单但有效的方法。第 2 章回顾了小波及其在统计学中的应用。这涉及紧支持小波基的构造、平方可积函数的小波变换以及在线性和非线性函数估计中的应用。对小波阈值化的文献进行了广泛的回顾，并得出了一些渐近结果。在第三章中，我们考虑多元时间序列中由于潜在确定性趋势的相似性而产生的依赖结构。平稳过程谱分析的结果扩展到确定性趋势
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