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)} $\\omega \\rho(\\|\\de-\\de_{0}\\|^{2}) +(1-\\omega)\\rho(\\|\\de - \\theta\\|^{2})$ 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 $.