Date of Award
Doctor of Philosophy
Daniel J. Nordman
This dissertation investigates and develops three different nonparametric likelihood methods for time series. For handling dependent data, current statistical methodology often relies on selecting parametric distributional models to accurately represent the data-generating process, which can be challenging. The three nonparametric likelihood methods considered are a blockwise empirical likelihood (EL), a block bootstrap, and a frequency domain bootstrap. Each method differs in form for building a nonparametric likelihood, but all methods involve "setting empirical probabilities" on observed data. An additional theme of this dissertation is the type or strength of the dependence in a stationary time process. The behavior of statistical methods can change dramatically between SRD and LRD cases, which complicates the development of appropriate resampling methods.
The dissertation consists of four chapters. Chatper 1 provides Introduction to explain the relationship about three manuscripts from Chatper 2, 3 and 4. Chapter 2 considers a new blockwise empirical likelihood (BEL) method for stationary, weakly dependent time processes, called a progressive block empirical likelihood (PBEL). Unlike the standard BEL originally proposed by Kitamura~(1997), the PBEL method does not require any block length selections. Because the performance of the standard BEL can depend critically on the block length choice, the PBEL method in contrast enjoys a type of robustness against block selection issues. Chapters 3 and 4 consider different bootstrap problems for stationary, linear time series which could exhibit LRD. Chapter 3 investigates the large-sample properties of a block bootstrap method for estimating the distribution of sample means. The results establish the validity of the block bootstrap under either LRD
or SRD. Additionally, for estimating the variance of a sample mean under LRD, explicit expressions are provided for the large-sample bias and variance of block bootstrap estimators along with formulas for the theoretically optimal block sizes under LRD. Perhaps surprisingly, optimal blocks become shorter in length as the strength of the LRD increases. Chapter 4 develops a frequency domain bootstrap (FDB) method for a problem involving Whittle estimation (Whittle, 1953), which is a popular technique for fitting parametric spectral density models to time series. For linear LRD time processes, the resulting Whittle estimators are known to have normal limit laws. However, convergence to normality can be slow under LRD and the finite-sample distributions of Whittle estimators tend to be asymmetric. As a remedy, the FDB method can be used for calibrating confidence intervals in place of a normal approximation.
Young Min Kim
Kim, Young Min, "On nonparametric likelihood methods for weakly and strongly dependent time series" (2012). Graduate Theses and Dissertations. 12365.