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Asymptotic distributions of likelihood ratios for overparametrized ARMA processes. (English) Zbl 0625.62074

Let \(e_ t\) be a Gaussian white noise and let \(y_ t\) be an ARMA(n,m) process generated by \[ y_ t+a_ 1y_{t-1}+...+a_ ny_{t-n}=e_ t+b_ 1e_{t-1}+...+b_ me_{t-m}. \] Assume that all roots of the polynomials \[ P(z)=1+a_ 1z+...+a_ nz^ n,\quad Q(z)=1+b_ 1z+...+b_ mz^ m \] lie in the set \(\{| z| >(1-\delta)^{- 1}\}\) where \(\delta >0\). The author considers two types of the log likelihood ratios for \(y_ t\). It is proved that their asymptotic distributions depend only on \(\delta\) and are invariant with respect to the parameters \(a_ j\), \(b_ k\). Some results of a simulation study are presented. The likelihood ratio test is proposed for model order reduction.
Reviewer: J.Anděl

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62F05 Asymptotic properties of parametric tests
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