Non-minimaxity of debiased shrinkage estimators

Abstract

AbstractWe consider the estimation of the p-variate normal mean of $$X\sim \mathcal {N}_p(\theta ,I)$$ X ∼ N p ( θ , 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 $$\Vert x\Vert ^2$$ ‖ x ‖ 2 and which is exactly equal to the unbiased estimator X for larger $$\Vert x\Vert ^2$$ ‖ x ‖ 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.