Simultaneous Estimation in a Restricted Linear Model

We consider a linear normal modelY=Xθ+ewithθverifying a linear restriction and the standard estimators θ(unrestricted MLE) andθ* (restricted MLE). We prove thatθ* is preferable to θusing 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 atθ* have more confidence than those centered at θ.