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Long-Memory Time Series: Theory and Methods (Wiley Series in Probability and Statistics)
by Wilfredo Palma
By Wilfredo Palma
Publisher: Wiley-Interscience
Number Of Pages: 304
Publication Date: 2007-03-30
ISBN-10 / ASIN: 0470114029
ISBN-13 / EAN: 9780470114025
Binding: Hardcover
During the last decades long-memory processes have evolved as a vital and important part of time series analysis. This book attempts to give an overview of the theory and methods developed to deal with long-range dependent data as well as describe some applications of these methodologies to real-life time series. The topics are systematically organized in a progressive manner, starting from foundations (the first three chapters), progressing to the analysis of methodological implications (the next six chapters), and finally extending to more complex long-range dependent data structures (the final three chapters).
Preface
Acronyms
1 Stationary Processes
1.1 Fundamental Concepts
1 . 1 . 1 Stationarity
1.1.2 Singularity and Regularity
1.1.3 Wold Decomposition Theorem
1.1.4 Causality
1.1.5 Invertibility
1.1.6 Best Linear Predictor
1.1.7 Szego-Kolmogorov Formula
1.1.8 Ergodicity
1.1.9 Martingales
1.1.10 Cumulants
1 . 1 . 1 1 Fractional Brownian Motion
1.1.12 Wavelets
1.2 Bibliographic Notes
Problems
15
16
2 State Space Systems
2.1 Introduction
2.1.1 Stability
2.1.2 Hankel Operator
2.1.3 Observability
2.1.4 Controllability
2.1.5 Minimality
2.2.1
2.2.2
2.2.3
2.3 Estimation of the State
2.3.1 State Predictor
2.3.2 State Filter
2.3.3 State Smoother
2.3.4 Missing Observations
2.3.5 Steady State System
2.3.6 Prediction of Future Observations
2.2 Representations of Linear Processes
State Space Form to Wold Decomposition
Wold Decomposition to State Space Form
Hankel Operator to State Space Form
2.4 Extensions
2.5 Bibliographic Notes
Problems
3 Long-Memory Processes
3.1 Defining Long Memory
3.1.1 Alternative Definitions
3.1.2 Extensions
3.2.1 Stationarity, Causality, and Invertibility
3.2.2 Infinite AR and MA Expansions
3.2.3 Spectral Density
3.2.4 Autocovariance Function
3.2.5 Sample Mean
3.2.6 Partial Autocorrelations
3.2.7 lllustrations
3.2.8 Approximation of Long-Memory Processes
3.2 AFWIMA Processes
3.3 Fractional Gaussian Noise
3.3.1 Sample Mean
3.4 Technical Lemmas
3.5 Bibliographic Notes
Problems
4 Estimation Methods
4.1 Maximum-Likelihood Estimation
4.1.1 Cholesky Decomposition Method
4.1.2 Durbin-Levinson Algorithm
4.1.3 Computation of Autocovariances
4.1.4 State Space Approach
4.2 Autoregressive Approximations
4.2.1 Haslett-Raftery Method
4.2.2 Beran Approach
4.2.3 A State Space Method
4.3 Moving-Average Approximations
4.4 Whittle Estimation
4.4.1 Other versions
4.4.2 Non-Gaussian Data
4.4.3 Semiparametric Methods
4.5.1 A Regression Method
4.5.2 Rescaled Range Method
4.5.3 Variance Plots
4.5.4 Detrended Fluctuation Analysis
4.5.5 A Wavelet-Based Method
4.5 Other Methods
4.6 Numerical Experiments
4.7 Bibliographic Notes
Problems
5 Asymptotic Theory
5.1 Notation and Definitions
5.2 Theorems
5.2.1 Consistency
5.2.2 Central Limit Theorem
5.2.3 Efficiency
5.3 Examples
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