On Truncation of Shrinkage Estimators in Simultaneous Estimation of Normal Means

Abstract

In estimating a multivariate normal mean θ = (θ₁, …, θ _k ) ^t under sum of squares error loss, it is well known that Stein estimators improve upon the usual estimator (in terms of expected loss) if k ≥ 3. The improvement obtained is significant, however, only if the θ _i are fairly close to the point towards which the Stein estimator shrinks. When extreme θ _i are likely (such as when the θ _i are thought to arise from a possibly heavy-tailed prior distribution), the standard Stein estimators may offer little improvement over the usual estimator. Stein (1981) proposed a limited translation Stein estimator to correct this deficiency. This estimator is analyzed herein for a number of heavy-tailed prior distributions. An adaptive version of the estimator is also discussed.

Key Words:

• James-Stein estimator
• Stein's truncation
• Heavy-tailed prior
• Adaptive estimator