Estimation of a parameter vector when some components are restricted

We consider the problem of estimating a p-dimensional parameter θ=(θ1,…,θp) when the observation is a p+k vector (X,U) where dim X=p and where U is a residual vector with dim U=k. The distributional assumption is that (X,U) has a spherically symmetric distribution around (θ,0). Two restrictions on the parameter θ are considered. First we assume that θi⩾0 for i=1,…,p and, secondly, we suppose that only a subset of these θi are nonnegative. For these two settings, we give a class of estimators δ(X,U)=δ0(X)+g(X)U′U which dominate, under the usual quadratic loss, a natural estimator δ0(X) which corresponds to the MLE in the normal case. Lastly, we deal with the situation where the parameter θ belongs to a cone C of Rp. We show that, under suitable condition, domination of the natural estimator adapted to this problem can be extended to a larger cone containing C and to any orthogonal transformation of this cone.