Statistical Reproduction of Orderings and Translation Subfamilies

Abstract

For a given family of distributions let $\varphi$ be a parameter which takes on different values on different subfamilies but is constant on each. The values of the parameter may be taken to define an ordering of the subfamilies. Various concepts of statistical reproduction of such an ordering are introduced. They generalize concepts such as unbiased estimators and tests. In the latter part of the paper it is assumed that in each subfamily all distributions may be obtained from a single one by translation of the random variable. As an application we consider testing for decrease in variability when the individuals tested may have different means.