Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Abstract

Let X ∼ f ( x − 2 ) and let (X) be the generalized Bayes estimator of with respect to a spherically symmetric prior, ( 2 ), for loss − 2 . We show that if (t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator 0 (X)=X under certain conditions on f ( ). The class of priors includes priors of the form 1 A+ 2 k for k p 2 − 1 and hence includes the fundamental harmonic prior 1 p−2 . The class of sampling distributions includes certain variance mixtures of normals and other functions f (t) of the form e − t and e − t+ (t ) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as (X) = 1 − ar(t) t X.

Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution

“…In the case of the normal mean vector, Zheng [1] gave conditions on modified Stein [2]'s harmonic priors for minimaxity and admissibility of the resultant Bayes estimators, and Strawderman [3] and Lin and Tsai [4] employed hierarchical priors to yield minimax Bayes estimators. Strawderman [3], Maruyama [5] and Fourdrinier and Strawderman [6] treated the problem of estimating a location vector in spherically symmetric distribution based on harmonic priors. Bayesian hierarchical models for estimation of the normal mean matrix were recently investigated by Berger et al [7] and Tsukuma [8,9].…”

Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance

“…Recently, [6,10,13,18] gave conditions for minimaxity of generalized Bayes estimators of the location vector of a spherically symmetric distribution under squared error loss. [13,18] consider the general spherical case whereas [10] consider the scale mixture of normals.…”

“…[13,18] consider the general spherical case whereas [10] consider the scale mixture of normals. The results in [10,13,18] do not cover the estimation problem considered here since the model sufficient statistics in (1.1) are (X, S), hence the corresponding posterior distribution will also depend on (X, S).…”

“…[13,18] consider the general spherical case whereas [10] consider the scale mixture of normals. The results in [10,13,18] do not cover the estimation problem considered here since the model sufficient statistics in (1.1) are (X, S), hence the corresponding posterior distribution will also depend on (X, S). The spherically symmetric distribution cases discussed in [10,13,16,18] do not have a scale parameter nor do they contain a "residual vector" (the spherically symmetric distribution analog of S) as in the case of spherically (or more generally elliptically) symmetric distributions discussed in [12].…”