Shrinkage estimators of the location parameter for certain spherically symmetric distributions

Abstract

spherical symmetry, quadratic loss, concave loss, location parameter, unknown scale,

Estimation under ℓ1-Symmetry

“…Similar results can be obtained following the line of Brandwein and Strawderman [2] and Brandwein et al [3] since the concavity of l permits a reduction to the case of the quadratic loss.…”

“…In [3], Brandwein et al use fundamentally the same assumptions except that, through restrictions to subclasses of spherically symmetric distributions, they can improve on the upper bound for b.…”

“…It is expressed in terms of a differential inequality involving the function g, that is 2(2 div g+ X." 2 g)+||g|| 2 [ 0 where, as usual, div g=; [3]. These authors need a superharmonicity condition on a function H connected to the shrinkage factor g for obtaining domination over X.…”

A Paradox Concerning Shrinkage Estimators: Should a Known Scale Parameter Be Replaced by an Estimated Value in the Shrinkage Factor?

“…The loss function is assumed to be &$&%& 2 . This problem has recently been considered by Brandwein and Strawderman [2,3], Brandwein et al [4], Cellier et al [7], and Cellier and Fourdrinier [6]. These papers study classes of estimators which improve on the usual article no.…”

On improved shrinkage estimators for concave loss

“…The main contribution is an improvement in shrinkage constants for minimax estimators over those in Brandwein andStrawderman (1980, 1991), and Brandwein et al (1993), particularly for variance mixtures of normals (and somewhat more generally), and for concave functions of squared error loss for which the derivative of the concave function is completely monotone (and somewhat more generally). For Baranchik-type estimators and for scale mixtures of multivariate normal distributions, we also show that our minimax improvements hold in dimension 3 which improves over the restriction that p ≥ 4 in the earlier papers.…”