Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis: A Frequency Domain Approach / Edition 1

by Gyïrgy Terdik

Overview

The object of the present work is a systematic statistical analysis of bilinear processes in the frequency domain. The first two chapters are devoted to the basic theory of nonlinear functions of stationary Gaussian processes, Hermite polynomials, cumulants and higher order spectra, multiple Wiener-Itô integrals and finally chaotic Wiener-Itô spectral representation of subordinated processes. There are two chapters for general nonlinear time series problems.

Product Details

ISBN-13:	9780387988726
Publisher:	Springer New York
Publication date:	07/30/1999
Series:	Lecture Notes in Statistics , #142
Edition description:	Softcover reprint of the original 1st ed. 1999
Pages:	270
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1 Foundations.- 1.1 Expectation of Nonlinear Functions of Gaussian Variables.- 1.2 Hermite Polynomials.- 1.2.1 Hermite polynomials of one Variable.- 1.2.2 Hermite polynomials of several variables.- 1.3 Cumulants.- 1.3.1 Definition of Cumulants.- 1.3.2 Basic Properties.- 1.4 Diagrams, and Moments and Cumulants for Gaussian Systems.- 1.4.1 Diagrams.- 1.4.2 Moments of Gaussian systems.- 1.4.3 Cumulants for Hermite polynomials.- 1.4.4 Products for Hermite polynomials.- 1.5 Stationary processes and spectra.- 1.5.1 Shastic spectral representation.- 1.5.2 Complex Gaussian system.- 1.5.3 Spectra.- 2 The Multiple Wiener-Itô Integral.- 2.1 Functions of Spaces $$ \overline {L_{\Phi }sub{n}} $$ and $$ \widetilde{{L_{\Phi }sub{n}}} $$.- 2.2 The multiple Wiener-Itô Integral of second order.- 2.2.1 Definition I.- 2.2.2 Definition II.- 2.2.3 Definition III.- 2.3 The multiple Wiener-Itô integral of order n.- 2.3.1 Properties.- 2.3.2 Diagram Formula.- 2.3.3 Fock space.- 2.3.4 Stratonovich integral in frequency domain and the Hu-Meyer formula.- 2.4 Chaotic representation of stationary processes.- 2.4.1 Subordinated functionals of Gaussian processes.- 2.4.2 Spectra for processes with Hermite degree-2.- 2.4.3 The process F (Xt).- 3 Stationary Bilinear Models.- 3.1 Definition of bilinear models.- 3.2 Identification of a bilinear model with scalar states.- 3.2.1 Multiple spectral representation and stationarity.- 3.2.2 Spectra.- 3.2.3 The necessary and sufficient condition for the existence of 2nth order moment, scalar case.- 3.3 Identification of bilinear processes, general case.- 3.3.1 State space form of lower triangular bilinear models.- 3.3.2 Vector valued bilinear model with scalar input.- 3.3.3 Spectra.- 3.3.4 Necessary and sufficient condition for the existence of 2nth order moments of the state process.- 3.4 Identification of multiple-bilinear models.- 3.4.1 Chaotic representation and stationarity.- 3.4.2 Spectra.- 3.5 State space realization.- 3.5.1 The bilinear realization problem.- 3.5.2 Realization of the Hermite degree-N homogeneous polynomial model.- 3.5.3 Minimal realizations.- 3.6 Some bilinear models of interest.- 3.6.1 Simple bilinear model.- 3.6.2 Hermite degree-2 bilinear model.- 3.7 Identification of GARCH(1,1) Model.- 3.7.1 Spectrum of the State Process.- 3.7.2 Spectrum of the square of the observations.- 3.7.3 Bispectrum of the state process.- 3.7.4 Bispectrum of the process Yt.- 3.7.5 Simulation.- 4 Non-Gaussian Estimation.- 4.1 Estimating a parameter for non-Gaussian data.- 4.2 Consistency and asymptotic variance of the estimate.- 4.3 Asymptotic normality of the estimate.- 4.4 Asymptotic variance in the case of linear processes.- 4.4.1 A worked example and simulations.- 5 Linearity Test.- 5.1 Quadratic predictor.- 5.1.1 Quadratic predictor for a simple bilinear model.- 5.2 The test statistics.- 5.3 Comments on computing the test statistics.- 5.4 Simulations and real data.- 5.4.1 Homogeneous bilinear realizable time series with Hermite degree-2.- 5.4.2 Results of simulations.- 6 Some Applications.- 6.1 Testing linearity.- 6.1.1 Geomagnetic Indices.- 6.1.2 Results of testing weak linearity for simulated data at WUECON.- 6.1.3 GARCH model fitting.- 6.2 Bilinear fitting.- 6.2.1 Parameter estimation for bilinear processes.- 6.2.2 Bilinear fitting for real data.- Appendix A Moments.- Appendix B Proofs for the Chapter Stationary Bilinear Models.- Appendix C Proofs for Section 3.6.1.- Appendix D Cumulants and Fourier Transforms for GARCH(1,1).- Appendix E Proofs for the Chapter Non-Gaussian Estimation.- E.0.1 Proof for Section 4.4.- Appendix F Proof for the Chapter Linearity Test.- References.