Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis: A Frequency Domain Approach - Softcover

Inhaltsangabe

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 Stochastic spectral representation.- 1.5.2 Complex Gaussian system.- 1.5.3 Spectra.- 2 The Multiple Wiener-Itô Integral.- 2.1 Functions of Spaces $$ \overline {L_{\Phi }^{n}} $$ and $$ \widetilde{{L_{\Phi }^{n}}} $$.- 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.

Die Inhaltsangabe kann sich auf eine andere Ausgabe dieses Titels beziehen.