Shrinkage estimates based on orthogonal decomposition of the sample space

Abstract

We show that for sufficiently large dimension the risk of the James-Stein estimate of a p-vector θ is reduced by decomposing the sample-space into orthogonal components and applying the James-Stein method within each. An example with two components is Lindley's estimate: for fixed p this has less risk than the James-Stein estimate if the -coefficient of variation of θ is not too large and the underlying sample size n is sufficiently large. An adaptive method of choosing how to decompose the sample-space is proposed. The effect of large n is also studied.

Keywords:

• James-Stein estimate
• multiple shrinkage
• multivariate normal
• orthogonal decomposition