Extended intrumental variable method: Recurrent algorithms (Q1290679)

The authors study recurrent estimation for multiple-input single-output ARMAX systems driven by stationary noise with rational spectral density. They propose an estimator of the form \[ \theta(t) = \alpha(t)\theta(t-1) + \beta(t)\varphi(t) , \] where \(\theta(t)\) is the new estimate of the unknown parameter at time \(t\), \(\varphi(t)\) is the input vector, \(\alpha(t)\) and \(\beta(t)\) are scalar coefficients to be determined. The coefficients \(\alpha(t)\) and \(\beta(t)\) are chosen to minimize a certain quadratic loss functional involving instrumental variables and prediction errors. The procedure admits a recurrent (in \(t\)) implementation. The selection of the instrumental variables is considered as well. Strong consistency is established under appropriate assumptions. The performance of the algorithm is studied on several example models.