On shrinkage estimation of a spherically symmetric distribution for balanced loss functions
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We consider the problem of estimating the mean vector θ of a d-dimensional spherically symmetric distributed X based on balanced loss functions of the forms: (i) ωρ(-_0^2) +(1-ω)ρ( - θ^2) and (ii) ℓ(ω - _0^2 +(1-ω) - θ^2), where δ_0 is a target estimator, and where ρ and ℓ are increasing and concave functions. For d≥ 4 and the target estimator δ_0(X)=X, we provide Baranchik-type estimators that dominate δ_0(X)=X and are minimax. The findings represent extensions of those of Marchand & Strawderman (<cit.>) in two directions: (a) from scale mixture of normals to the spherical class of distributions with Lebesgue densities and (b) from completely monotone to concave ρ' and ℓ'.
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