On estimation with weighted balanced-type loss function

Abstract

For estimating an unknown parameter θ , we introduce and motivate the use of the balanced-type loss function: L ω , δ 0 ( θ , δ ) = ω q ( θ ) ( δ - δ 0 ) 2 + ( 1 - ω ) q ( θ ) ( δ - θ ) 2 , where 0 ⩽ ω ⩽ 1 , q ( θ ) is a positive weight function, and δ 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 ω , δ 0 may be inferred directly from corresponding results for weighted squared error loss (i.e., ω = 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 θ under loss L ω , δ 0 with δ 0 being the maximum-likelihood estimator of θ .