Sequential point estimation of parameters in a threshold AR(1) model

We show that if an appropriate stopping rule is used to determine the sample size when estimating the parameters in a stationary and ergodic threshold AR(1) model, then the sequential least-squares estimator is asymptotically risk efficient. The stopping rule is also shown to be asymptotically efficient. Furthermore, non-linear renewal theory is used to obtain the limit distribution of appropriately normalized stopping rule and a second-order expansion for the expected sample size. A central result here is the rate of decay of lower-tail probability of average of stationary, geometrically β-mixing sequences.