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

Document Type : Research Paper


1 Bushehr University of Medical Sciences ,Bushehr, Iran

2 Department of Statistics, Velayat University, Velayat, Iran

3 Department of Statistics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran


In this paper, we discuss the two-stage and the modified two-stage procedures for the estimation of the threshold autoregressive parameter in a first-order threshold autoregressive model (${\rm TAR(1)}$). 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.


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