Inadmissible estimators of normal quantiles and two-sample problems with additional information

Abstract

We consider estimation problem of a normal quantile μ+ησ. For the scale invariant squared error loss and unrestricted values of the population mean and standard deviation μ and σ, [13] established the inadmissibility of the MRE estimator for η≠0. In this paper, we explore: (i) the impact of the loss with the study of scale invariant absolute value loss, and (ii) situations where there is a parameter space restriction of a lower bounded mean μ. We establish

(i) the inadmissibility of the MRE estimator of μ+ησ; η≠0; under scale invariant absolute value loss;

(ii) the inadmissibility of the Generalized Bayes estimator of μ+ησ; η>0; under scale invariant squared error loss, associated with the prior measure 1_(0,∞)(μ)1_(0,∞)(σ) which represents the truncation of the usual non-informative prior measure onto the restricted parameter space.

Both of these results are obtained through a conditional risk analysis and may be viewed as extensions of [13]. Finally, we provide further applications to two-sample problems under the presence of the additional information of ordered means.