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: , where 0[less-than-or-equals, slant][omega][less-than-or-equals, slant]1, q([theta]) is a positive weight function, and [delta]0 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],[delta]0 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],[delta]0 with [delta]0 being the maximum-likelihood estimator of [theta].