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

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)} $omega rho(|de-de_{0}|^{2}) +(1-omega)rho(|de - theta|^{2})$ and {bf (ii)} $ellleft(omega |de - de_{0}|^{2} +(1-omega)|de - theta|^{2}right)$, where $delta_0$ is a target estimator, and where $rho$ and $ell$ are increasing and concave functions. For $dgeq 4$ 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 $rho$ and $ell$.