On the Relative Behavior of Point Estimators Based on a Decomposition of Mean Absolute Error

Abstract

**Please note that the full text is embargoed** ABSTRACT: Let [see pdf for notation] be a family of probability density functions indexed by the parameter [see pdf for notation]. We assume at least one of the [see pdf for notation] is unknown. Based on a random sample of size n from [see pdf for notation], let [see pdf for notation] be two point estimators of the real-valued function [see pdf for notation], where [see pdf for notation] are specified constants, if any. When comparing [see pdf for notation] and [see pdf for notation], it is quite common to examine the ratio of their respective average precisions usually measured by either mean squared error, [see pdf for notation], or mean absolute error, [see pdf for notation], where [see pdf for notation]. If, for example, [see pdf for notation] for some w0' then 02 is said to be more mean squared efficient than [see pdf for notation] at [see pdf for notation]. However, the numerical value of such a ratio provides very limited insight into the actual relative behavior of the two competing estimators. We, therefore, propose a twofold technique for comparing [see pdf for notation] and which essentially determines (a) the "odds" in favor of [see pdf for notation] being closer to [see pdf for notation] than is [see pdf for notation] and (b) the average closeness of [see pdf for notation] to [see pdf for notation] not only when [see pdf for notation] is closer to [see pdf for notation] than is [see pdf for notation] but also when it is not. Closeness to [see pdf for notation] is measured through an absolute error loss function: [see pdf for notation]. Furthermore, joint consideration of these two concepts is shown to provide a basis for determining which of the two estimators, [see pdf for notation] or [see pdf for notation], is preferred in a given situation. An application of these results will be made with regard to the comparison of estimators of certain reliability characteristics in the exponential failure model.