Estimating normal means with symmetric gain functions (Q1099526)

When \(X_ 1,...,X_ n\) are independent and identically distributed normal random variables with unknown mean \(\theta\) and known variance, the estimation problem of \(\theta\) is considered. It is shown that the decision chosen by Bayes rule with a prior density of double exponential type and some symmetric gain function is evaluated in comparison with the sufficient statistic \(\bar X=(X_ 1+...+X_ n)/n\). The above is extended to the case of estimating several means. The previous results by the same author, Stochastic Processes Appl. 14, 267-277 (1983; Zbl 0499.62013); Suppl. Issues Stat. and Decis. 1, 47-59 (1984; Zbl 0558.62025), are involved as special cases.