On shrinkage estimation for balanced loss functions

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Abstract

The estimation of a multivariate mean θ is considered under natural modifications of balanced loss functions of the form: (i) ωρ(δδ02)+(1ω)ρ(δθ2), and (ii) ωδδ02+(1ω)δθ2, where δ0 is a target estimator of γ(θ). After briefly reviewing known results for original balanced loss with identity ρ or , we provide, for increasing and concave ρ and 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 .

AMS 2010 subject classifications

62F10
62J07
62C15
62C20

Keywords

Balanced loss
Concave loss
Dominance
Multivariate normal
Scale mixture of normals
Shrinkage estimation

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