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(‖δ−δ0‖2)+(1−ω)ρ(‖δ−θ‖2) and (ii) ℓω‖δ−δ0‖2+(1−ω)‖δ−θ‖2, where δ0 is a target estimator, and where ρ and ℓ are increasing and concave functions. For d≥4 and the target estimator δ0(X)=X, we provide Baranchik-type estimators that dominate δ0(X)=X and are minimax. The findings represent extensions of those of Marchand & Strawderman (2020) 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′.