On the inevitability of a paradox in shrinkage estimation for scale mixtures of normals

Let (X,U) be a random vector with spherically symmetric distribution about (θ,0) where dim X=dim θ=p and dim U=dim 0=k (for p≥3). Consider estimation of θ with loss ∥θ-d∥². Fourdrinier and Strawderman (J. Multivariate Anal. 59 (1996) 109) considered two classes of James-Stein type estimators δ_α^a (X,U)=(1-a[(U′U)^α/X′X])X for α=0 and 1. The case α=0 is referred to as a "classical" James-Stein estimator and the case α=1 as a "robust" James-Stein type estimator since α=0 corresponds to the original James-Stein estimator in the normal case with known variance. In contrast, the case α=1 corresponds to an estimator which, for a=(p-2)/(k+2), simultaneously and uniformly dominates X for all spherically symmetric distributions. Fourdrinier and Strawderman showed that, for certain spherically symmetric distributions, the optimal (a=(p-2)/(k+2)) James-Stein estimator for α=1 simultaneously dominates all James-Stein estimators with α=0. They term this situation a paradox since an estimated value of σ² in the shrinkage constant leads to an estimator which improves over the entire class of estimators which use the known value of σ² in the shrinkage constant. We show in this paper that this paradox is inevitable whenever the underlying distribution is a nondegenerate mixture of normal distributions and the dimension, k, of the residual vector U is sufficiently large. We also calculate the "critical dimension" k₀ for a variety of examples including the Student-t.