The loss of efficiency estimating linear functions under restrictions

This article is motivated by the problem of estimating contrast in a one-wa y ANOVA model with restrictions in the parameter vector. We prove that when the restrictions are given by a tree order or a simple order the MLE of so me contrast has greater MSE than the unrestricted estimator. A similar beha viour of the MLE is exhibited in a general restricted setting given by a mu ltivariate normal distribution with mean vector constrained to belong to a circular cone. The approach we use focuses on the central direction of the cones. These directions appear to have the greater MSE when the dimension o f the restricted cone is big enough.