Estimation of location parameters for spherically symmetric distributions

AbstractEstimation of the location parameters of a p×1 random vector X̲ with a spherically symmetric distribution is considered under quadratic loss. The conditions of Brandwein and Strawderman [Ann. Statist. 19(1991) 1639–1650] under which estimators of the form X̲+ag̲(X̲) dominate X̲are (i) ∥g̲∥2/2⩽-h⩽-▽∘g̲, where -h is superharmonic, (ii) Eθ̲[R2h(V̲)] is nonincreasing in R, where V̲ has a uniform distribution in the sphere centered at θ̲ with a radius R, and (iii) 0<a⩽1/[pE0̲(∥X̲∥-2)]. In this paper, we not only drop their condition (ii) to show the dominance of X̲+ag̲(X̲) over X̲, but also obtain a new bound for a which is sometimes better than that obtained by Brandwein and Strawderman. Specifically, the new bound of a is 0<a<[μ1/(p2μ-1)][1-(p-1)μ1/(pμ-1μ2)]-1 with μi=E0̲(∥X̲∥i) for i=-1,1,2. The generalization to concave loss functions is also considered. Additionally, we investigate estimators of the location parameters when the scale is unknown and the observation contains a residual vector