Observed data based estimator and both observed data and prior information based estimator under asymmetric losses

Abstract

In Bayesian approach of statistical analyses we incorporate the prior information about the parameter of the model with the observed data. This prior information is in the form of a prior distribution of the parameter. If the prior information is available as a constant value of the parameter rather than its' prior distribution, the Bayesian approach cannot be pursued. However, there are estimation methods that incorporate such prior information with the observed data. The expectation is that the incorporation of such additional information in the estimation process would result in a better estimator than that based on the observed data alone. In some cases this may be true, but in many other cases the risk of worse consequences cannot be ruled out. This paper studies the performance of the observed data based unrestricted estimator (UE), and both observed data and prior information based preliminary test estimator (PTE) of the univariate normal mean under the linex loss function. The risk functions of both UE and PTE are derived. The moment generating function (MGF) of PTE is derived which turns out to be a component of the risk function. From the MGF the first two moments of PTE are obtained and found to be identical to that obtained by using different approaches in Khan and Saleh (2001) and Zellner (1986). Under the linex loss criterion the performance of the PTE is compared with that of UE. It is revealed that if the uncertain non-sample prior information about the value of the mean is not too far from its true value, PTE outperforms UE.