Non-minimaxity of debiased shrinkage estimators

Abstract

We consider the estimation of the $p$-variate normal mean of $X\sim N_p(\theta ,I)$ under the quadratic loss function. We investigate the decision theoretic properties of debiased shrinkage estimator, the estimator which shrinks towards the origin for smaller $\Vert x{\Vert }^2$ and which is exactly equal to the unbiased estimator $X$ for larger $\Vert x{\Vert }^2$. Such debiased shrinkage estimator seems superior to the unbiased estimator $X$, which implies minimaxity. However we show that it is not minimax under mild conditions.