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)} $\ell\left(\omega \|\de - \de_{0}\|^{2} +(1-\omega)\|\de - \theta\|^{2}\right)$, where $\delta_0$ is a target estimator, and where $\rho$ and $\ell$ are increasing and concave functions. For $d\geq 4$ 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 $\ell'$.