Stationary process

Abstract: Preface 1 INTRODUCTION 1.1 Examples of Time Series 1.2 Objectives of Time Series Analysis 1.3 Some Simple Time Series Models 1.3.3 A General Approach to Time Series Modelling 1.4 Stationary Models and the Autocorrelation Function 1.4.1 The Sample Autocorrelation Function 1.4.2 A Model for the Lake Huron Data 1.5 Estimation and Elimination of Trend and Seasonal Components 1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality 1.5.2 Estimation and Elimination of Both Trend and Seasonality 1.6 Testing the Estimated Noise Sequence 1.7 Problems 2 STATIONARY PROCESSES 2.1 Basic Properties 2.2 Linear Processes 2.3 Introduction to ARMA Processes 2.4 Properties of the Sample Mean and Autocorrelation Function 2.4.2 Estimation of $\gamma(\cdot)$ and $\rho(\cdot)$ 2.5 Forecasting Stationary Time Series 2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past Values 2.6 The Wold Decomposition 1.7 Problems 3 ARMA MODELS 3.1 ARMA($p,q$) Processes 3.2 The ACF and PACF of an ARMA$(p,q)$ Process 3.2.1 Calculation of the ACVF 3.2.2 The Autocorrelation Function 3.2.3 The Partial Autocorrelation Function 3.3 Forecasting ARMA Processes 1.7 Problems 4 SPECTRAL ANALYSIS 4.1 Spectral Densities 4.2 The Periodogram 4.3 Time-Invariant Linear Filters 4.4 The Spectral Density of an ARMA Process 1.7 Problems 5 MODELLING AND PREDICTION WITH ARMA PROCESSES 5.1 Preliminary Estimation 5.1.1 Yule-Walker Estimation 5.1.3 The Innovations Algorithm 5.1.4 The Hannan-Rissanen Algorithm 5.2 Maximum Likelihood Estimation 5.3 Diagnostic Checking 5.3.1 The Graph of $\t=1,\ldots,n\ 5.3.2 The Sample ACF of the Residuals

Abstract: This article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations. Previously, a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m-dimensional joint distribution of m consecutive observations, where m is fixed. This procedure has been generalized by constructing a “blocks of blocks” resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole (i.e., infinite-dimensional) joint distribution of the stationary sequence of observations. These methods share the construction of resampling blocks of observations to form a pseudo-time series, so that the statistic of interest may be recalculated based on the resampled data set. But in the context of applying this method to stationary data, it is natural...

Abstract: Preliminaries: Time Series and Spectra, Summary of Vector Space Geometry, Some Probability Notations and Properties. Models for Spectral Analysis - The Univariate Case: The Wiener Theory of Spectral Analysis, Stationary and Weakly Stationary Stochastic Processes, The Spectral Representation for Weakly Stationary Stochastic Processes - A Special Case, The General Spectral Representation for Weakly Stationary Processes, The Discrete and Continuous Components of the Process, Physical Realizations of the Different Kinds of Spectra, The Real Spectral Representation, Ergodicity and the Connection between the Wiener and Stationary Process Theories, Statistical Estimation of the Autocovariance and the Mean Ergodic Theorem. Sampling, Aliasing, and Discrete-Time Models: Sampling and the Aliasing Problem, The Spectral Model for Discrete-Time Series Linear Filters - General Properties with Applications to Continuous-Time Processes: Linear Filters, Combining Linear Filters, Inverting Linear Filters, Nonstationary Processes Generated by Time Varying Linear Filters. Multivariate Spectral Models and Their Applications: The Spectrum of a Multivariate Time Series-Wiener Theory, Multivariate Weakly Stationary Stochastic Processes, Linear Filters for Multivariate Time Series, The Bivariate Spectral Parameters, Their Interpretations and Uses. The Multivariate Spectral Parameters, Their Interpretations and Uses Digital Filters: General Properties of Digital Filters, The Effect of Finite Data Length, Digital Filters with Finitely Many Nonzero Weights, Digital Filters Obtained by Combining Simple Filters, Filters with Gapped Weights and Results Concerning the Filtering of Series with Polynomial Trends. Finite Parameter Models, Linear Prediction and Real-Time Filtering: Moving Averages, Autoregressive Processes, The Linear Prediction Problem, Mixed Autoregressive-Moving Average Processes and Recursive Prediction, Linear Filtering in Real Time. The Distribution Theory of Spectral Estimates with Applications to Statistical Inference: Distribution of the Finite Fourier Transform and the Periodogram. Distribution Theory for Univariate Spectral Estimators, Distribution Theory for Multivariate Spectral Estimators with Applications to Statistical Inference. Sampling Properties of Spectral Estimates, Experimental Design and Spectral Computations, Properties of Spectral Estimators and the Selection of Spectral Windows, Experimental Design, Methods for Computing Spectral Estimators, Data Processing Problems and Techniques.