Estimation of location parameters for spherically symmetric distributions

Estimation of the location parameters of a p × 1 random vector X ̲ 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 X ̲ + a g ̲ ( X ̲ ) dominate X ̲ are (i) ∥ g ̲ ∥ 2 / 2 ⩽ - h ⩽ - ▽ ∘ g ̲ , where - h is superharmonic, (ii) E θ ̲ [ R 2 h ( V ̲ ) ] is nonincreasing in R , where V ̲ has a uniform distribution in the sphere centered at θ ̲ with a radius R , and (iii) 0 < a ⩽ 1 / [ pE 0 ̲ ( ∥ X ̲ ∥ - 2 ) ] . In this paper, we not only drop their condition (ii) to show the dominance of X ̲ + a g ̲ ( X ̲ ) over X ̲ , 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 < [ μ 1 / ( p 2 μ - 1 ) ] [ 1 - ( p - 1 ) μ 1 / ( p μ - 1 μ 2 ) ] - 1 with μ i = E 0 ̲ ( ∥ X ̲ ∥ 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.