Estimation of a normal mean relative to balanced loss functions

Let X-1,..., X-n be a random sample from a normal distribution with mean theta and variance sigma(2). The problem is to estimate theta with Zellner's (1994) balanced loss function, L-B((theta) over cap,theta) = omega/n Sigma(1)(n) (X-i - (theta) over cap)(2) + (1 - omega)(theta-(theta) over cap)(2) where 0 < omega < 1. It is shown that the sample mean (X) over bar, is admissible. More generally, we investigate the admissibility of estimators of the form a (X) over bar + b under L-B((theta) over cap,theta). We also consider the weighted balanced loss function, L-W ((theta) over cap, theta) = omegaq(theta) Sigma(1)(n) (X-i-(theta) over cap)(2) + (1 - omega)q(theta)(theta -(theta) over cap)(2), where q(theta) is any positive function of theta, and the class of admissible linear estimators is obtained under such loss with q(theta) = e(theta).