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

Abstract

From an observable \((X,U)\) in \(\mathbb R^p \times \mathbb R^k\), we consider estimation of an unknown location parameter \(\theta \in \mathbb R^p\) under two distributional settings: the density of \((X,U)\) is spherically symmetric with an unknown scale parameter \(\sigma \) and is ellipically symmetric with an unknown covariance matrix \(\Sigma \). Evaluation of estimators of \(\theta \) is made under the classical invariant losses \(\Vert d - \theta \Vert ^2 / \sigma ^2\) and \((d - \theta )^t \Sigma ^{-1} (d - \theta )\) as well as two respective data based losses \(\Vert d - \theta \Vert ^2 / \Vert U\Vert ^2\) and \((d - \theta )^t S^{-1} (d - \theta )\) where \(\Vert U\Vert ^2\) estimates \(\sigma ^2\) while \(S\) estimates \(\Sigma \). 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 \(1 / \sigma ^2\)) 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 \(X\) 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 \(t\) and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well.