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) $\omega \rho (\Vert \delta -{\delta }_0{\Vert }^2)+(1-\omega )\rho (\Vert \delta -\theta {\Vert }^2)$, and (ii) $\ell \left(\omega \Vert \delta -{\delta }_0{\Vert }^2+(1-\omega )\Vert \delta -\theta {\Vert }^2\right)$, 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

15 pages