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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.

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)