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.