James-Stein Type Estimators Shrinking towards Projection Vector When the Norm is Restricted to an Interval

Abstract

Consider the problem of estimating a $p{\times}1$ 수식 이미지 mean vector ${\theta}(p-q{\geq}3)$ 수식 이미지, $q=rank(P_V)$ 수식 이미지 with a projection matrix $P_v$ 수식 이미지 under the quadratic loss, based on a sample $X_1$ 수식 이미지, $X_2$ 수식 이미지, ${\cdots}$ 수식 이미지, $X_n$ 수식 이미지. We find a James-Stein type decision rule which shrinks towards projection vector when the underlying distribution is that of a variance mixture of normals and when the norm ${\parallel}{\theta}-P_V{\theta}{\parallel}$ 수식 이미지 is restricted to a known interval, where $P_V$ 수식 이미지 is an idempotent and projection matrix and rank $(P_V)=q$ 수식 이미지. In this case, we characterize a minimal complete class within the class of James-Stein type decision rules. We also characterize the subclass of James-Stein type decision rules that dominate the sample mean