By Dr. T. Subba Rao, Dr. M. M. Gabr (auth.)

The idea of time sequence versions has been good constructed over the past thirt,y years. either the frequenc.y area and time area ways were common within the research of linear time sequence versions. notwithstanding. many actual phenomena can't be properly represented through linear versions; as a result the need of nonlinear types and better order spectra. lately a couple of nonlinear types were proposed. during this monograph we limit realization to 1 specific nonlinear version. referred to as the "bilinear model". the main attention-grabbing function of this type of version is that its moment order covariance research is ve~ just like that for a linear version. This demonstrates the significance of upper order covariance research for nonlinear types. For bilinear versions it's also attainable to procure analytic expressions for covariances. spectra. and so on. that are usually tricky to procure for different proposed nonlinear types. Estimation of bispectrum and its use within the building of assessments for linearit,y and symmetry also are mentioned. the entire tools are illustrated with simulated and genuine info. the 1st writer wish to recognize the convenience he obtained within the education of this monograph from offering a sequence of lectures regarding bilinear types on the college of Bielefeld. Ecole Normale Superieure. college of Paris (South) and the Mathematisch Cen trum. Ams terdam.

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**Example text**

To illustrate this. Lii and Rosenblatt (1982). have considered the two representations. namely. 3) Xt = 6V t - 5V t _1 + Vt - 2 • Yt = 3V t - 7V t _l + 2V t _2 (1. 4) where {V t } are independent. with zero mean and variance one. 3) have roots 2 and 3. 4) have roots i and 3. and the two processes have the same spectral density function. If the independent random variables {V t } are exponentially distributed. then the marginal distributions of Xt and Yt are different. Now consider the linear representation Xt = L j=_co a.

It can be shown (Brillinger and Rosenblatt, 1967a, 1967b; Van Ness, 1966) that although I(Wl'wZ) is an asymptotically unbiased estimate of f(Wl'WZ)' it is not a consistent estimate of f(wl'wz)' To obtain a consistent estimate, I(Wl'WZ) has to be "smoothed", as in the second order case. 4) = MZ where KM(6 1,6z) such that MZ/N + Ko(M6 1,M6 z ), and M, the window parameter, is chosen 0 as M+~, N + ~. Since f(Wl'WZ) and f(Wl'WZ) are complex valued functions, we can write f(Wl'WZ) = r(Wl'WZ) + i q(Wl'WZ), f(wpwz) = r (Wl'wZ) +; q(wj>wz) Then the mean and the variance of f(Wl'WZ) can be defined by var(f(wl'wZ» , , Z , , Z = E\f(Wl'WZ) - E f(Wl>WZ) \ = E[r(Wl'WZ) - E(r(wl'wz)] , , + E[q(Wl'WZ'> - E(q(wl'wz)] Z The mean square error is defined by = var(f(Wl'WZ» where b(Wl'WZ)' the t:Jias, ;s given by + \b(Wl'WZ) I Z 40 We need the following definition to obtain an expression for the bias.

The number of observations generated for all the four series is 500. The theoretical and the estimated spectral densities far 3eries C and D are plotted in Figs. 4 respectively. We note that the spectral shapes of the non-linear series D, C and D are similar to the spectral shape of Series A. 4. TZ) is given by A 1 N-y C(T1 ,TZ) = N L t=l _ _ (Xt-X) (X t + -X) (X t + -X). 2) The function I(wpwz) is called the "third order periodogram". It can be shown (Brillinger and Rosenblatt, 1967a, 1967b; Van Ness, 1966) that although I(Wl'wZ) is an asymptotically unbiased estimate of f(Wl'WZ)' it is not a consistent estimate of f(wl'wz)' To obtain a consistent estimate, I(Wl'WZ) has to be "smoothed", as in the second order case.