Recursive estimation of a drifted autoregressive parameter

Suppose the X0,...., Xn are observations of a one-dimensional stochastic dynamic process described by autoregression equations when the autoregressive parameter is drifted with time, i.e. it is some function of time: ¿0,...., ¿n, with ¿k = ¿ (k/n). The function ¿(t) is assumed to belong a priori to a predetermined nonparametric class of functions satisfying the Lipschitz smoothness condition. At each time point t those observations are accessible which have been obtained during the preceding time interval. A recursive algorithm is proposed to estimate ¿??t??. Under some conditions on the model, we derive the rate of convergence of the proposed estimator when the frequencyof observations n tends to infinity.