Shrinkage Estimators under Spherical Symmetry for the General Linear Model

This paper is primarily concerned with extending the results of Brandwein and Strawderman in the usual canonical setting of a general linear model when sampling from a spherically symmetric distribution. When the location parameter belongs to a proper linear subspace of the sampling space, we give an unbiased estimator of the difference of the risks between the least squares estimator φ0 and a general shrinkage estimator φ = φ0 − ∥X − φ0 ∥2 · g ∘ φ0. We obtain a general condition of domination for φ over φ0 which is weaker than that of Brandwein and Strawderman. We do not need any superharmonicity condition on g. Our results are valid for general quadratic loss.