The book provides the foundation of time series methods, including linear filters and a geometric approach to prediction. The important paradigm of ARMA models is studied in-depth, as well as frequency domain methods. Entropy and other information theoretic notions are introduced, with applications to time series modeling.
Auteur
Tucker S. McElroy is Senior Time Series Mathematical Statistician at the U.S. Census Bureau, where he has contributed to developing time series research and software for the last 15 years. He has published more than 80 papers and is a recipient of the Arthur S. Flemming award (2011).
Dimitris N. Politis is Distinguished Professor of Mathematics at the University of California at San Diego, where he is also serving as Associate Director of the Halicioglu Data Science Institute. He has co-authored two research monographs and more than 100 journal papers. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (2009-2011) and is Co-Editor of the Journal of Time Series Analysis.
Texte du rabat
Time Series: A First Course with Bootstrap Starter provides an introductory course on time series analysis that satisfies the triptych of (i) mathematical completeness, (ii) computational illustration and implementation, and (iii) conciseness and accessibility to upper-level undergraduate and M.S. students. Basic theoretical results are presented in a mathematically convincing way, and the methods of data analysis are developed through examples and exercises parsed in R. A student with a basic course in mathematical statistics will learn both how to analyze time series and how to interpret the results.
The book provides the foundation of time series methods, including linear filters and a geometric approach to prediction. The important paradigm of ARMA models is studied in-depth, as well as frequency domain methods. Entropy and other information theoretic notions are introduced, with applications to time series modeling. The second half of the book focuses on statistical inference, the fitting of time series models, as well as computational facets of forecasting. Many time series of interest are nonlinear in which case classical inference methods can fail, but bootstrap methods may come to the rescue. Distinctive features of the book are the emphasis on geometric notions and the frequency domain, the discussion of entropy maximization, and a thorough treatment of recent computer-intensive methods for time series such as subsampling and the bootstrap. There are more than 600 exercises, half of which involve R coding and/or data analysis. Supplements include a website with 12 key data sets and all R code for the book's examples, as well as the solutions to exercises.
Contenu
The Probabilistic Structure of Time Series Random Vectors Time Series and Stochastic Processes Marginals and Strict Stationarity Autocovariance and Weak Stationarity Illustrations of Stochastic Processes Three Examples of White Noise Overview Exercises
Trends, Seasonality, and Filtering Nonparametric Smoothing Linear Filters and Linear Time Series Some Common Types of Filters Trends Seasonality Trend and Seasonality Together Integrated Processes Overview Exercises
The Geometry of Random Variables Vector Space Geometry and Inner Products L2(; P;F): The Space of Random Variables with Finite Second Moment Hilbert Space Geometry Projection in Hilbert Space Prediction of Time Series Linear Prediction of Time Series Orthonormal Sets and Infinite Projection Projection of Signals Overview Exercises
ARMA Models with White Noise Residuals Definition of the ARMA Recursion Difference Equations Stationarity and Causality of the AR(1) Causality of ARMA Processes Invertibility of ARMA Processes The Autocovariance Generating Function Computing ARMA Autocovariances via the MA Representation Recursive Computation of ARMA Autocovariances Overview Exercises
Time Series in the Frequency Domain The Spectral Density Filtering in the Frequency Domain Inverse Autocovariances Spectral Representation of Toeplitz Covariance Matrices Partial Autocorrelations Application to Model Identification Overview Exercises
The Spectral Representation The Herglotz Theorem The Discrete Fourier Transform The Spectral Representation Optimal Filtering Kolmogorov's Formula The Wold Decomposition Spectral Approximation and the Cepstrum Overview Exercises
Information and Entropy Introduction Events and Information Sets Maximum Entropy Distributions Entropy in Time Series Markov Time Series Modeling Time Series via Entropy Relative Entropy and Kullback-Leibler Discrepancy Overview Exercises
Statistical Estimation Weak Correlation and Weak Dependence The Sample Mean CLT for Weakly Dependent Time Series Estimating Serial Correlation The Sample Autocovariance Spectral Means Statistical Properties of the Periodogram Spectral Density Estimation Refinements of Spectral Analysis Overview Exercises
Fitting Time Series Models MA Model Identification EXP Model Identification AR Model Identification Optimal Prediction Estimators Relative Entropy Minimization Computation of Optimal Predictors Computation of the Gaussian Likelihood Model Evaluation Model Parsimony and Information Criteria Model Comparisons Iterative Forecasting Applications to Imputation and Signal Extraction Overview Exercises
Nonlinear Time Series Analysis Types of Nonlinearity The Generalized Linear Process The ARCH Model The GARCH Model The Bi-spectral Density Volatility Filtering Overview Exercises
The Bootstrap Sampling Distributions of Statistics Parameters as Functionals and Monte Carlo The Plug-in Principle and the Bootstrap Model-based Bootstrap and Residuals Sieve Bootstraps Time Frequency Toggle Bootstrap Subsampling Block Bootstrap Methods Overview Exercises
A. Probability Probability Spaces Random Variables Expectation and Variance Joint Distributions The Normal Distribution Exercises
B. Mathematical Statistics Data Sampling Distributions Estimation Inference Con_dence Intervals Hypothesis Testing Exercises
C. Asymptotics Convergence Topologies Convergence Results for Random Variables Asymptotic Distributions Central Limit Theory for Time Series Exercises
D. Fourier Series Complex Random Variables Trigonometric Polynomials
E. Stieltjes Integration Deterministic Integration Stochastic Integration