Time Series: Theory and Methods
Springer Science & Business Media, Nov 11, 2013 - Mathematics - 520 pages
We have attempted in this book to give a systematic account of linear time series models and their application to the modelling and prediction of data collected sequentially in time. The aim is to provide specific techniques for handling data and at the same time to provide a thorough understanding of the mathematical basis for the techniques. Both time and frequency domain methods are discussed but the book is written in such a way that either approach could be emphasized. The book is intended to be a text for graduate students in statistics, mathematics, engineering, and the natural or social sciences. It has been used both at the M. S. level, emphasizing the more practical aspects of modelling, and at the Ph. D. level, where the detailed mathematical derivations of the deeper results can be included. Distinctive features of the book are the extensive use of elementary Hilbert space methods and recursive prediction techniques based on innovations, use of the exact Gaussian likelihood and AIC for inference, a thorough treatment of the asymptotic behavior of the maximum likelihood estimators of the coefficients of univariate ARMA models, extensive illustrations of the tech niques by means of numerical examples, and a large number of problems for the reader. The companion diskette contains programs written for the IBM PC, which can be used to apply the methods described in the text.
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29 Hilbert Space Isomorphisms
Stationary ARMA Processes
44 Spectral Densities and ARMA Processes
49 Inversion Formulae
103 Asymptotic Properties of the Periodogram
6 The Cross Spectrum
122 Transfer Function Modelling
124 Long Memory Processes
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absolutely summable algorithm applying approximation AR(p ARMA process ARMA(p,q asymptotic autocorrelation function autocovariance function autoregressive best linear predictor Cauchy Cauchy–Schwarz inequality characteristic function closed subspace coefficients complex-valued compute converges Corollary covariance function covariance matrix denote difference equations differenced dZ(v element Example Figure finite follows Fourier frequency function F Gaussian Hence Hilbert space independent inner product inner-product space integer lags maximum likelihood estimators mean squared error mean zero moving average multivariate normal non-negative definite non-zero observations obtain orthogonal-increment process orthonormal parameters partial autocorrelation periodogram polynomial Problem process defined program PEST projection theorem PROOF properties Proposition random variables random vector real-valued recursively Remark sample autocorrelation function satisfying Section sequence ſº spectral density spectral distribution function spectral representation stationary process stationary solution stationary time series uncorrelated unique values variance white noise