On shrinkage estimation for balanced loss functions

04/05/2019

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by   Eric Marchand, et al.

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The estimation of a multivariate mean θ is considered under natural modifications of balanced loss function of the form: (i) ω ρ(δ-δ_0^2) + (1-ω) ρ(δ-θ^2) , and (ii) ℓ( ω δ-δ_0^2 + (1-ω) δ-θ^2 ) , where δ_0 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 δ_0(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