two-stage and modified two-stage estimation in threshold first-order autoregressive process

In this paper, we discuss the two-stage and the modified twostage procedures for the estimation of the threshold autoregressive parameter in a first-order threshold autoregressive model (tar(1)). this is motivated by the problem of finding a final sample size when the sample size is unknown in advance. for this purpose, a two-stage stopping variable and a class of modified two-stage stopping variables are proposed. afterward, we prove the significant properties of the procedures, including asymptotic efficiency and asymptotic risk efficiency for the point estimation based on least-squares estimators. to illustrate this theory, comprehensive monte carlo simulation studies is conducted to observe the significant properties of the procedures. furthermore, the performance of procedures based on yule-walker estimators is investigated and the results are compared in practice that confirm our theoretical results. finally, real-time-series data is studied to demonstrate the application of the procedures.