On shrinkage estimation for balanced loss functions

Abstract: The estimation of a multivariate mean $\theta$ is considered under natural modifications of balanced loss function of the form: (i) $\omega \, \rho(|\delta-\delta_0|^2) + (1-\omega) \, \rho(|\delta-\theta|^2) $, and (ii) $\ell \left( \omega \, |\delta-\delta_0|^2 + (1-\omega) \, |\delta-\theta|^2 \right)\,$, where $\delta_0$ is a target estimator of $\gamma(\theta)$. After briefly reviewing known results for original balanced loss with identity $\rho$ or $\ell$, we provide, for increasing and concave $\rho$ and $\ell$ which also satisfy a completely monotone property, Baranchik-type estimators of $\theta$ which dominate the benchmark $\delta_0(X)=X$ 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 $\rho$ or $\ell