Estimation of a parameter vector when some components are restricted

We consider the problem of estimating a p-dimensional parameter [theta]=([theta]1,...,[theta]p) 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 ([theta],0). Two restrictions on the parameter [theta] are considered. First we assume that [theta]i[greater-or-equal, slanted]0 for i=1,...,p and, secondly, we suppose that only a subset of these [theta]i are nonnegative. For these two settings, we give a class of estimators [delta](X,U)=[delta]0(X)+g(X)U'U which dominate, under the usual quadratic loss, a natural estimator [delta]0(X) which corresponds to the MLE in the normal case. Lastly, we deal with the situation where the parameter [theta] belongs to a cone of . We show that, under suitable condition, domination of the natural estimator adapted to this problem can be extended to a larger cone containing and to any orthogonal transformation of this cone.