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(\mathbf{i}\mathbf{i})$\ell\left(\omega |\de - \de_{0}|^{2} +(1-\omega)|\de - \theta|^{2}\right)$,where$\delta_0$ is a target estimator, and where $\rho$ and $\mathrm{\ell }$ 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 ${\rho }^{\mathrm{\prime }}$ and ${\mathrm{\ell }}^{\mathrm{\prime }}$.