Data based loss estimation of the mean of a spherical distribution with a residual vector

In the canonical setting of the general linear model, we are concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. More precisely, from an observable ( X , U ) in ℝ^p ×ℝ^k having a density of the form 1 / σ ^p+k f ( ( ‖x- θ‖ ^2 + ‖u‖ ^2 / σ ^2 ) ) where θ and σ are both unknown, we consider general estimators φ (X,‖ U‖ ^2) of θ under two losses: the usual quadratic loss ‖φ (X,‖ U‖ ^2) - θ‖ ^2 and the data-based loss ‖φ (X,‖ U‖ ^2) - θ‖ ^2 / ‖ U‖ ^2 . Then, for each loss, we compare, through a squared error risk, their unbiased loss estimator δ _0(X,‖ U‖ ^2) with a general alternative loss estimator δ (X,‖ U‖ ^2) . Thanks to the new Stein type identity in Fourdrinier and Strawderman (Metrika 78(4):461–484, 2015), we provide an unbiased estimator of the risk difference between δ (X,‖ U‖ ^2) and δ _0(X,‖ U‖ ^2) , which gives rise to a sufficient domination condition of δ (X,‖ U‖ ^2) over δ _0(X,‖ U‖ ^2) . Minimax estimators of Baranchik form illustrate the theory. It is found that the distributional assumptions and dimensional conditions on the residual vector U are weaker when the databased loss is used.