Robust minimax Stein estimation under invariant data-based loss for spherically and elliptically symmetric distributions

From an observable in , we consider estimation of an unknown location parameter under two distributional settings: the density of is spherically symmetric with an unknown scale parameter and is ellipically symmetric with an unknown covariance matrix . Evaluation of estimators of is made under the classical invariant losses and as well as two respective data based losses and where estimates while estimates . We provide new Stein and Stein-Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of ) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24-39, 2003). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well.