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

TL;DR

This paper develops improved shrinkage estimators for the mean of spherically symmetric distributions under balanced loss functions, extending previous results to broader distribution classes and concave loss functions.

Contribution

It introduces Baranchik-type estimators that dominate the usual estimator and are minimax for higher dimensions, extending prior work to more general distributions and loss functions.

Findings

Proposed estimators dominate the usual estimator for dimensions d ≥ 4.

Extensions from scale mixtures of normals to broader spherical distributions.

Generalization from monotone to concave 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 $\ell'$.