The prescribed precision estimators of the autoregression parameter using the generalized least square method

A sequential estimator is proposed for the autoregression parameter of first-order (AR(1)), which is constructed on the basis of a generalized least square method (GLSM) using a special choice of the weight coefficients in the sum of residual squares. Under some natural requirements on the noise distribution function, this is the prescribed precision estimator in the sense that it provides the unknown parameter estimation with any fixed square average accuracy at the moment of termination of the observation. In contrast to the sequential least square estimator, our estimator has the important property of uniform asymptotic normality with respect to the parameter on the whole axis. Using this result one can show that the sequential least square estimator is asymptotically optimal in the minimax sense for the power loss function, in a wide class of sequential and nonsequential procedures.