On estimation with weighted balanced-type loss function

For estimating an unknown parameter theta, we introduce and motivate the use of the balanced-type loss function: L-omega,L-delta o (theta, delta) = omega q(theta)(delta-delta(0))(2) + (1 - omega)q(theta)(delta - theta)(2), where 0 <= omega <= 1, q(theta) is a positive weight function, and delta(o) is a general target estimator. Developments and various examples are given with regards to the issues of admissibility, dominance, Bayesianity, and minimaxity. In many cases, as in Dey et al. [1999. On estimation with balanced loss functions. Statist. Probab. Lett. 45, 97-101], we show that results for loss L-omega,L-delta o may be inferred directly from corresponding results for weighted squared error loss (i.e., omega = 0). Specific issues related to constrained parameter spaces, which include the choice of the target estimator, are addressed. Finally, we derive minimax estimators of a bounded normal mean theta under loss L-omega,L-delta o with delta(o) being the maximum-likelihood estimator of theta. (C) 2005 Elsevier B.V. All rights reserved.