On shrinkage estimation for balanced loss functions

The estimation of a multivariate mean θ is considered under natural modifications of balanced loss functions of the form: (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 of γ(θ). After briefly reviewing known results for original balanced loss with identity ρ or ℓ, we provide, for increasing and concave ρ and ℓ which also satisfy a completely monotone property, Baranchik-type estimators of θ which dominate the benchmark δ₀(X)=X for X either distributed as multivariate normal or as a scale mixture of normals. Implications are given with respect to model robustness and simultaneous dominance with respect to either ρ or ℓ.