On a criterion for the selection of models for stationary time series
Consider a stationary time series \(x_ t\) with a spectral density f. One desires to fit ''the best model'' from a family with spectral densities \(g_{\theta}\), \(\theta \in R_ p\). Denote \(k(\omega,\theta)=\sigma_ g^{-2} 2\pi g_{\theta}(\omega)\), where \(\sigma^ 2_ g\) is the innovation variance. let \[ \phi =\partial k^{-1}/\partial \theta,\quad \psi =\partial^ 2k^{-1}/\partial \theta^ 2,\quad I(\omega)=(2\pi T)^{-1}| \sum^{T}_{s=1}x_ se^{i\omega (s-1)}|^ 2. \] Let \({\hat \theta}_ T\) be a strongly consistent estimator of \(\theta\) based on \(x_ 1,...,x_ T\). The proposed criterion is \[ \Delta_ T({\hat \theta}_ T)+T^{-1} tr({\hat \Omega}^{-1} {\hat \Sigma}), \] where \[ \Delta_ T(\theta)=(2\pi /T)\sum^{T-1}_{t=0}I(\omega_ t)k^{-1}(\omega_ t,\theta),\quad \omega_ t=2\pi t/T, \] \[ {\hat \Omega}=(2\pi /T)\sum^{T-1}_{t=0}\psi (\omega_ t,{\hat \theta}_ T)I(\omega_ t), \] \[ {\hat \Sigma}=(4\pi^ 2/T)\sum^{T-1}_{t=0} \phi(\omega_ t,{\hat \theta}_ T)\phi '(\omega_ t,{\hat \theta}_ T)I^ 2(\omega_ t). \]
scientific article; zbMATH DE number 3980298
scientific article