Fixed Accuracy Estimation of an Autoregressive Parameter

For a first order non-explosive autoregressive process with unknown parameter β ∈ -1, 1, it is shown that if data are collected according to a particular stopping rule, the least squares estimator of β is asymptotically normally distributed uniformly in β. In the case of normal residuals, the stopping rule may be interpreted as sampling until the observed Fisher information reaches a preassigned level. The situation is contrasted with the fixed sample size case, where the estimator has a non-normal unconditional limiting distribution when |β| = 1.