Recursive estimation of a drifted autoregressive parameter

Abstract

Suppose the $X_0,\dots, X_n$ 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: $\theta_0,\dots, \theta_n$, with $\theta_k = \theta(k/n)$. The function $\theta(t)$ 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 $\theta(t)$.Under some conditions on the model,we derive the rate of convergence of the proposed estimator when the frequencyof observations $n$ tends to infinity.