Unbiased Estimation of Location and Scale Parameters

Abstract

It is well-known that there is a close connection between linear functionals on an appropriate Banach space and unbiased estimators. In Section 2 we prove some results concerning unbiased estimation of location and scale parameters. As application of these results we consider the case of Cauchy density with unknown location [scale] but known scale [location] parameter. We show that there exists no unbiased estimator for the location parameter, and none with finite variance for the scale parameters. If the Cauchy density involves both location and scale parameters, then it is shown that neither of these parameters has an unbiased estimator. Some information about other parametric functions is also given. The present results for the location parameter case were obtained previously by H. Pollard; we are grateful to Professor Kiefer for informing us of Pollard's work.