A New Generalized Estimator for AR(1) Model Which Improves MLE Uniformly

For the first-order autoregressive model, Ochi (Journal of Time Series Analysis 4:57–67, 1983) introduced a generalized estimator of the coefficient $$\alpha $$ α with two constants $$c_1$$ c 1 and $$c_2$$ c 2 , which includes Daniels’ estimator, least-squares estimator, and Durbin’s estimator. From Fujikoshi and Ochi (Annals of the Institute of Statistical Mathematics 36:119–128, 1984), compared with a third-order approximated estimator of maximum likelihood estimator, it was shown that the modified maximum likelihood estimator (MLE) is better than the modified Ochi’s estimator in the third-order sense if we modify the both estimators to be “third-order asymptotically median unbiased”. In this paper, we propose a new estimator when the $$c_1$$ c 1 and $$c_2$$ c 2 depend on $$\alpha $$ α , i.e., $$c_1(\alpha )$$ c 1 ( α ) and $$c_2(\alpha )$$ c 2 ( α ) . Then we show that it improves the MLE uniformly in the sense of the third-order mean square error “without bias-adjustments”. Because $$\alpha $$ α is unknown, the feasible estimator with $$c_1(\hat{\alpha })$$ c 1 ( α ^ ) and $$c_2(\hat{\alpha })$$ c 2 ( α ^ ) is proposed with the replacement of $$\alpha $$ α bt a consistent estimator $$\hat{\alpha }$$ α ^ .