A minimax regret estimator of a normal mean after preliminary test

Summary

This paper considers the problem of estimating a normal mean from the point of view of the estimation after preliminary test of significance. But our point of view is different from the usual one. The difference is interpretation about a null hypothesis. Let$\bar X$ denote the sample mean based on a sample of sizen from a normal population with unknown mean μ and known varianceσ ². We consider the estimator that assumes the value$\omega \bar X$ when$\left| {\bar X} \right|{{< C\sigma } \mathord{\left/ {\vphantom {{< C\sigma } {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}$ and the value$\bar X$ when$\left| {\bar X} \right|{{ \geqq C\sigma } \mathord{\left/ {\vphantom {{ \geqq C\sigma } {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}$ where ω is a real number such that 0≤ω≤1 andC is some positive constant. We prove the existence of ω, satisfying the minimax regret criterion and make a numerical comparison among estimators by using the mean square error as a criterion of goodness of estimators.