Project Euclid

This paper gives a class of minimum $L_2$-distance estimators of the autoregression parameter in the first-order autoregression model when the errors have an unknown symmetric distribution. Within the class an asymptotically efficient estimator is exhibited. The asymptotic efficiency of this estimator relative to the least-squares estimator is the same as that of a certain signed rank estimator relative to the sample mean in the one sample location model. The paper also discusses goodness-of-fit tests for testing for symmetry and for a specified error distribution.