On shrinkage estimation of a spherically symmetric distribution for balanced loss functions

Abstract

We consider the problem of estimating the mean vector $\theta$ of a $d$-dimensional spherically symmetric distributed $X$ based on balanced loss functions of the forms: {\bf (i)} $ωρ(\|\de-\de_{0}\|^{2}) +(1-ω)ρ(\|\de - θ\|^{2})$ and {\bf (ii)} $\ell\left(ω\|\de - \de_{0}\|^{2} +(1-ω)\|\de - θ\|^{2}\right)$, where ${\delta }_0$ is a target estimator, and where $\rho$ and $\ell$ are increasing and concave functions. For $d\ge 4$ and the target estimator ${\delta }_0(X)=X$, we provide Baranchik-type estimators that dominate ${\delta }_0(X)=X$ and are minimax. The findings represent extensions of those of Marchand \& Strawderman (\cite{ms2020}) in two directions: {\bf (a)} from scale mixture of normals to the spherical class of distributions with Lebesgue densities and {\bf (b)} from completely monotone to concave ${\rho }^{\prime }$ and ${\ell }^{\prime }$.