Covariance structure associated with an equality between two general ridge estimators

In the Gauss–Markov model, this paper derives a necessary and sufficient condition under which two general ridge estimators coincide with each other. The condition is given as a structure of the dispersion matrix of the error term. Since the class of estimators considered here contains linear unbiased estimators such as the ordinary least squares estimator and the best linear unbiased estimator, our result can be viewed as a generalization of the well known theorems on the equality between these two estimators, which have been fully studied in the literature. Two related problems are also considered: equality between two residual sums of squares, and classification of dispersion matrices by a perturbation approach.