LetX1,…,Xnbe a random sample from a normal distribution with mean θ and variance σ2. The problem is to estimate θ with Zellner's (1994) balanced loss function,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$L_B \left( {\hat \theta ,\theta } \right) = \frac{\omega }{n}\sum {_1^n \left( {X_i - \hat \theta } \right)^2 } + \left( {1 - w} \right)\left( {\theta ,\hat \theta } \right)^2 $$
\end{document}% MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$\bar X$$
\end{document}% MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$\alpha \bar X + b$$
\end{document}% MathType!End!2!1! under\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$L_B \left( {\hat \theta ,\theta } \right)$$
\end{document}% MathType!End!2!1!. We also consider the weighted balanced loss function,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$L_W \left( {\hat \theta ,\theta } \right) = \omega q\left( \theta \right)\frac{{\sum {_1^n \left( {X_i - \hat \theta } \right)^2 } }}{n} + \left( {1 - w} \right)q\left( \theta \right)\left( {\theta ,\hat \theta } \right)^2 $$
\end{document}% MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =eθ.
