## Abstract

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 σ^{2} in the shrinkage constant leads to an estimator which improves over the entire class of estimators which use the known value of σ^{2} 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" k_{0} for a variety of examples including the Student-t.

Original language | English (US) |
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Pages (from-to) | 37-51 |

Number of pages | 15 |

Journal | Journal of Statistical Planning and Inference |

Volume | 121 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2004 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics

## Keywords

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