Characterization of estimators uniformly shrinking on subspaces

Shrinkage factors play an important role in the behaviour of biased estimators. In this paper, we first show that the only way to have bounded shrinkage factors on a subspace is to shrink uniformly on this subspace. Then, we characterize regressions on components that shrink uniformly on the subspaces spanned by their associated weight vectors. We show that this problem is equivalent to solving a set of linear equations involving two different projectors. We define a class of matrices whose eigen decompositions give the solution.