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Applications of innovation representations in time series analysis. (English) Zbl 0673.62085
Probability and statistics, Essays in honor of Franklin A. Graybill, 61-84 (1988).
Summary: [For the entire collection see Zbl 0667.00023.]
If $$\{X_ t\}$$ is a zero-mean second order process, it is known that the best linear mean-square predictor $$\hat X_{n+1}$$ of $$X_{n+1}$$ based on $$X_ 1,...,X_ n$$ is expressible in terms of the innovations $$(X_ j-\hat X_ j)$$, $$j=1,...,n$$ as $$\sum^{n}_{j=1}\theta_{nj}(X_{n+1-j}-\hat X_{n+1-j})$$, where the coefficients $$\theta_{nj}$$ and the mean squared errors $$v_ n=E(X_{n+1}-\hat X_{n+1})^ 2$$ can be found recursively from the covariance function of $$\{X_ t\}$$. If $$\{X_ t\}$$ is an ARMA (p,q) process defined by the equations, $$\phi (B)X_ t=\theta (B)Z_ t$$, then application of the recursions to the process $$\{\phi (B)X_ t\}$$ gives the more compact representation, $\hat X_{n+1}=\phi_ 1X_ n+...+\phi_ pX_{n-p}+\sum^{q}_{j=1}\theta_{nj}(X_{n+\quad 1- j}-\hat X_{n+1-j}),\quad n\geq \max (p,q).$ Applications of these representations to inference problems for time series are investigated.

##### MSC:
 62M20 Inference from stochastic processes and prediction 62M15 Inference from stochastic processes and spectral analysis 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)