Class of Modified Two-Stage Procedure in a Autoregressive Process

In this paper, we first discuss the class of modified two-stage procedure for estimation of the autoregressive parameter in a first-order autoregressive model (AR(1)). We prove the significant properties of the modified two-stage procedure, including asymptotic efficiency, asymptotic risk efficiency, and asymptotic consistency for the point and the interval estimation based on least-squares estimators. Then, the introduced class is generalized to the p-order autoregressive model (AR(p)) and is checked for their asymptotic properties. Also, we conduct comprehensive Monte Carlo simulation studies to test the properties of the proposed procedure based on least-squares estimators and Yule-Walker estimators in practice. Finally, a real-time series is provided to investigate the applicability of the class of modified two-stage variables.