Author	Terdik, Gyรถrgy. author
Title	Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis [electronic resource] : A Frequency Domain Approach / by Gyรถrgy Terdik
Imprint	New York, NY : Springer New York : Imprint: Springer, 1999
Connect to	http://dx.doi.org/10.1007/978-1-4612-1552-3
Descript	XV, 270 p. 25 illus. online resource

"Ninety percent of inspiration is perspiration. " [31] The Wiener approach to nonlinear stochastic systems [146] permits the representation of single-valued systems with memory for which a small perยญ turbation of the input produces a small perturbation of the output. The Wiener functional series representation contains many transfer functions to describe entirely the input-output connections. Although, theoretically, these representations are elegant, in practice it is not feasible to estimate all the finite-order transfer functions (or the kernels) from a finite samยญ ple. One of the most important classes of stochastic systems, especially from a statistical point of view, is the case when all the transfer functions are determined by finitely many parameters. Therefore, one has to seek a finite-parameter nonlinear model which can adequately represent nonยญ linearity in a series. Among the special classes of nonlinear models that have been studied are the bilinear processes, which have found applicaยญ tions both in econometrics and control theory; see, for example, Granger and Andersen [43] and Ruberti, et al. [4]. These bilinear processes are deยญ fined to be linear in both input and output only, when either the input or output are fixed. The bilinear model was introduced by Granger and Andersen [43] and Subba Rao [118], [119]. Terdik [126] gave the solution of xii a lower triangular bilinear model in terms of multiple Wiener-It(') integrals and gave a sufficient condition for the second order stationarity. An imporยญ tant

1 Foundations -- 1.1 Expectation of Nonlinear Functions of Gaussian Variables -- 1.2 Hermite Polynomials -- 1.3 Cumulants -- 1.4 Diagrams, and Moments and Cumulants for Gaussian Systems -- 1.5 Stationary processes and spectra -- 2 The Multiple Wiener-Itรด Integral -- 2.1 Functions of Spaces $$ \overline {L_{\Phi }̂{n}} $$ and $$ \widetilde{{L_{\Phi }̂{n}}} $$ -- 2.2 The multiple Wiener-Itรด Integral of second order -- 2.3 The multiple Wiener-Itรด integral of order n -- 2.4 Chaotic representation of stationary processes -- 3 Stationary Bilinear Models -- 3.1 Definition of bilinear models -- 3.2 Identification of a bilinear model with scalar states -- 3.3 Identification of bilinear processes, general case -- 3.4 Identification of multiple-bilinear models -- 3.5 State space realization -- 3.6 Some bilinear models of interest -- 3.7 Identification of GARCH(1,1) Model -- 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 -- 5 Linearity Test -- 5.1 Quadratic predictor -- 5.2 The test statistics -- 5.3 Comments on computing the test statistics -- 5.4 Simulations and real data -- 6 Some Applications -- 6.1 Testing linearity -- 6.2 Bilinear fitting -- 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