On shrinkage estimation for balanced loss functions

The estimation of a multivariate mean theta is considered under natural modifications of balanced loss functions of the form: (i) omega rho(parallel to delta - delta(0)parallel to(2)) + (1 - omega) rho(parallel to delta - theta parallel to(2)), and (ii) l (omega parallel to delta - delta(0)parallel to(2) + (1 - omega) parallel to delta - theta parallel to(2)), where delta(0) is a target estimator of gamma(theta). After briefly reviewing known results for original balanced loss with identity rho or l, we provide, for increasing and concave rho and l which also satisfy a completely monotone property, Baranchik-type estimators of theta which dominate the benchmark delta(0)(X) = X for X either distributed as multivariate normal or as a scale mixture of normals. Implications are given with respect to model robustness and simultaneous dominance with respect to either rho or l. (C) 2019 Elsevier Inc. All rights reserved.