Asymptotic statistical properties of AR spectral estimators for processes with mixed spectra

Copyright © 2002 IEEEThe influence of a point spectrum on large sample statistics of the autoregressive (AR) spectral estimator is addressed. In particular, the asymptotic distributions of the AR coefficients, the innovations variance, and the spectral density estimator of a finite-order AR(p) model to a mixed spectrum process are presented. Various asymptotic results regarding AR modeling of a regular process with a continuous spectrum are arrived at as special cases of the results for the mixed spectrum setting. Finally, numerical simulations are performed to verify the analytical resultsSoon-Sen Lau, P. J. Sherman and L. B. Whit

Statistical signal processing for mechanical systems

Random processes such as temperature and acoustic noise are found in all types of mechanical systems. Knowledge of these processes can lead to improved design and detection methods related to faulty operation. The goal of this dissertation is to contribute to the knowledge base of such processes. Specifically, we address statistical signal processing methods that are appropriate and consistent relative to the physics of these systems. Two generic problems associated with random signal measurements from mechanical systems are addressed.;Random processes associated with mechanical systems usually have complex spectral structure containing both continuous and line spectral components. Accordingly, they are called mixed random processes. One problem addressed is to use variability related to families of spectral estimators for a mixed random process to better characterize its spectral information. We show that tones are a significant source of bias and variability of families of spectral estimators. Expressions for estimating statistical and arithmetic variability of three common families of spectral estimators are provided. An important and immediate application of these results is tone detection.;We also address the statistical problem of estimating the bandwidth parameter of a Gauss-Markov process from a realization of fixed and finite duration at selectable sampling interval. The motivation is that continuous-time processes are often sampled at a rate far higher than their underlying dynamics. It is commonly assumed a faster sample rate is better. But in many real world situations, such as in adaptive feedback control schemes design, short time changes demand only limited time being utilized. Thus this problem is investigated. The bias and variance expressions of the parameter estimator are derived with a second order expansion. Three sample rate regions---finite, large and very large ones, corresponding to substantial, gradual, and very slight variance drop, are quantitatively identified. Guidelines in choosing sampling rate based on estimator performance requirement are provided.;The results are used to characterize the stochastic structure of the sound pressure process from an engine cooling fan with and without mock engine, and to perform a hypothesis test for deciding whether a design change has a significant effect on the sound

Linearization Methods in Time Series Analysis

In this dissertation, we propose a set of computationally efficient methods based on approximating/representing nonlinear processes by linear ones, so-called linearization. Firstly, a linearization method is introduced for estimating the multiple frequencies in sinusoidal processes. It utilizes a regularized autoregressive (AR) approximation, which can be regarded as a "large p - small n" approach in a time series context. An appealing property of regularized AR is that it avoids a model selection step and allows for an efficient updating of the frequency estimates whenever new observations are obtained. The theoretical analysis shows that the regularized AR frequency estimates are consistent and asymptotically normally distributed. Secondly, a sieve bootstrap scheme is proposed using the linear representation of generalized autoregressive conditional heteroscedastic (GARCH) models to construct prediction intervals (PIs) for the returns and volatilities. Our method is simple, fast and distribution-free, while providing sharp and well-calibrated PIs. A similar linear bootstrap scheme can also be used for diagnostic testing. Thirdly, we introduce a robust lagrange multiplier (LM) test, which utilizes either the bootstrap or permutation procedure to obtain critical values, for detecting GARCH effects. We justify that both bootstrap and permutation LM tests are consistent. Intensive numerical studies indicate that the proposed resampling algorithms significantly improve the size and power of the LM test in both skewed and heavy-tailed processes. Moreover, fourthly, we introduce a nonparametric trend test in the presence of GARCH effects (NT-GARCH) based on heteroscedastic ANOVA. Our empirical evidence show that NT-GARCH can effectively detect non-monotonic trends under GARCH, especially in the presence of irregular seasonal components. We suggest to apply the bootstrap procedure for both selecting the window length and finding critical values. The newly proposed methods are illustrated by applications to astronomical data, to foreign currency exchange rates as well as to water and air pollution data. Finally, the dissertation is concluded by an outlook on further extensions of linearization methods, e.g., in model order selection and change point detection