Bootstrap and inference for some linear time series models

In this dissertation, three papers are presented which cover topics such as comparing two different bootstrap approximations of the sampling distribution of an M-estimator for an ARMA model, use of the moving block bootstrap with a linear time series process with heavy tailed innovations, and comparing an estimate of the index variation as applied to the original data of an AR(p) model as well as to the estimated innovations of the AR(p) model. Chapter 1 introduces the reader to the various topics involved as well as to needed terminology and to previous literature on the subject matter. Chapters 2, 3, and 4 cover the various topics of interest and Chapter 5 summarizes the results in all three middle chapters and presents possible future research topics. The following paragraphs give a quick overview of the material in Chapters 2, 3, and 4.; Kreiss and Franke (1992) proposed bootstrapping a linear approximation to the M-estimator in ARMA models. In Chapter 2, it is argued that it may be better to apply the bootstrap principle directly to the M-estimator itself. A number of simulation results are presented to compare the two procedures for estimating the sampling distribution of an M-estimator. The theoretical asymptotic validity of the standard bootstrap applied to the M-estimator is established.; In Chapter 3 we study the moving block bootstrap approximation to the sampling distribution of a least squares estimator of the index of regular variation in a linear process with innovations satisfying a standard tail regularity condition. Sufficient conditions are obtained for the asymptotic validity of the procedure. A number of simulation studies are included to examine its finite sample behavior.; From Chapter 3 it is known that a linear process with innovations satisfying a standard tail regularity condition inherits a similar distribution as the innovations themselves with the same index parameter. Hence, in Chapter 4 the variability and stability of the least squares estimator of the index parameter as proposed by Datta and McCormick (1997) is compared when applied to an AR(p) model with innovations satisfying a standard tail regularity condition and to the estimated innovations of the AR(p) model. Consistency of the least squares estimator applied to the estimated innovations is established. Also, simulation results are presented which show empirically that the latter estimator has less variability and is more stable than the least squares estimator applied to the linear process itself.