We consider a linear normal model Y=X%+e with % verifying a linear restriction and the standard estimators % (unrestricted MLE) and %* (restricted MLE). We prove that %* is preferable to % using a new and strong criterion which implies the domination under other usual criteria; in particular it is proven that the standard simultaneous confidence intervals centered at %* have more confidence than those centered at % . 1997 Academic Press
1. Introduction
We consider the estimation of the parameter vector in a linear model Y=X%+e, e ^Nk (0, _ 2 I ), where X is a k_p complete range matrix and % is restricted to belong to a polyhedral cone ^/R p . The unrestricted maximum likelihood estimator (MLE) % for % is the standard estimator % =(X$X) &1 X$Y, % ^Np (%, _ 2 (X$X) &1 ), and the restricted MLE, %*, is the projection of % on ^for the metric given by 7=X$X : %*=P 7 (% Â^). (See Robertson et al. [11] for this and other topics in restricted inference.) This general framework includes as a particular case the problem of comparing the means of independent normal populations, which is the standard model in the context of restricted inference, where % is the vector of the population means, % is the vector of sample means, and 7 is diagonal.
We are interested in comparing the performance of % and %*. It is well known that when % is estimated globally, the restricted MLE is better than the unrestricted MLE under the quadratic error loss; however, using other loss functions we could obtain the opposite conclusion.