Weighted least squares and prediction (Q1201369)

This paper introduces a weighted least-squares (WLS) method for estimation of the parameters in (pseudo) linear time-invariant regression models. It is proved that the WLS parameter estimator proposed behaves like the unweighted least-squares estimator in the usual parameter estimation case, and also that the former yields prediction errors having the properties of the prediction errors corresponding to the stochastic- gradient parameter estimator. The main tool of which is made use in the proofs is a weighted law of large numbers for vector-valued martingales. Owing to the bounds on quadratic means shown to hold true in the case of the WLS estimation, the solution to the adaptive tracking problem derived by using this estimator is shown, under weak conditions, to possess good properties such as consistency and almost-surely guaranteed convergence rates.