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.