The loss of efficiency estimating linear functions under restrictions

This article is motivated by the problem of estimating contrast in a one-way 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 some contrast has greater MSE than the unrestricted estimator. A similar behaviour of the MLE is exhibited in a general restricted setting given by a multivariate 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 of the restricted cone is big enough.