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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)