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

Dominique Fourdrinier, Eric Marchand, William E. Strawderman

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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 δαa (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.

Original languageEnglish (US)
Pages (from-to)37-51
Number of pages15
JournalJournal of Statistical Planning and Inference
Issue number1
StatePublished - Mar 1 2004

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


  • Laplace transforms
  • Location parameter
  • Quadratic loss
  • Scale mixtures of normals
  • Shrinkage estimation


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