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 $${\mathbb {R}}^p \times {\mathbb {R}}^k$$ R p × R k having a density of the form $$1 / \sigma ^{p+k} \, f \! \left( \big ( \Vert {\textbf{x}}- \varvec{\theta }\Vert ^2 + \Vert {\textbf{u}}\Vert ^2 / \sigma ^2 \big ) \right) $$ 1 / σ p + k f ( ‖ x - θ ‖ 2 + ‖ u ‖ 2 / σ 2 ) where $$\varvec{\theta }$$ θ and $$\sigma $$ σ are both unknown, we consider general estimators $$ \varphi (X,\Vert U\Vert ^2) $$ φ ( X , ‖ U ‖ 2 ) of $$\varvec{\theta }$$ θ under two losses: the usual quadratic loss $$\Vert \varphi (X,\Vert U\Vert ^2) - \varvec{\theta }\Vert ^2$$ ‖ φ ( X , ‖ U ‖ 2 ) - θ ‖ 2 and the data-based loss $$\Vert \varphi (X,\Vert U\Vert ^2) - \varvec{\theta }\Vert ^2 / \Vert U\Vert ^2$$ ‖ φ ( X , ‖ U ‖ 2 ) - θ ‖ 2 / ‖ U ‖ 2 . Then, for each loss, we compare, through a squared error risk, their unbiased loss estimator $$\delta _0(X,\Vert U\Vert ^2)$$ δ 0 ( X , ‖ U ‖ 2 ) with a general alternative loss estimator $$\delta (X,\Vert U\Vert ^2)$$ 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 $$\delta (X,\Vert U\Vert ^2)$$ δ ( X , ‖ U ‖ 2 ) and $$\delta _0(X,\Vert U\Vert ^2)$$ δ 0 ( X , ‖ U ‖ 2 ) , which gives rise to a sufficient domination condition of $$\delta (X,\Vert U\Vert ^2)$$ δ ( X , ‖ U ‖ 2 ) over $$\delta _0(X,\Vert U\Vert ^2)$$ 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.