On shrinkage estimation for balanced loss functions

Éric Marchand, William E. Strawderman

Research output: Contribution to journalArticle

Abstract

The estimation of a multivariate mean θ is considered under natural modifications of balanced loss functions of the form: (i) ωρ(‖δ−δ02)+(1−ω)ρ(‖δ−θ‖2), and (ii) ℓω‖δ−δ02+(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.

Original languageEnglish (US)
Article number104558
JournalJournal of Multivariate Analysis
Volume175
DOIs
StatePublished - Jan 2020

Fingerprint

Balanced Loss Function
Shrinkage Estimation
Completely Monotone
Model Robustness
Estimator
Scale Mixture
Multivariate Normal
Benchmark
Target
Loss function
Shrinkage estimation

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

Keywords

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

Cite this

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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(‖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.",
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On shrinkage estimation for balanced loss functions. / Marchand, Éric; Strawderman, William E.

In: Journal of Multivariate Analysis, Vol. 175, 104558, 01.2020.

Research output: Contribution to journalArticle

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AU - Marchand, Éric

AU - Strawderman, William E.

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N2 - 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.

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