Bounds on AREs for Restricted Classes of Distributions Defined Via Tail-Orderings

Abstract

It is shown that large classes of asymptotic relative efficiencies (AREs) are isotonic with respect to various partial orderings on the heaviness of tails of symmetric distributions. The orderings include those of van Zwet (1964), Lawrence (1975), Barlow and Proschan (1975), and a new one that generalizes all three. Characterizations in terms of these orderings are given for many familiar families of distributions with restricted tail and central behavior. By restricting attention to such distributions, finite bounds are obtained for AREs such as that of some robust estimates to the sample mean, which could be unbounded otherwise. Similar results are shown to hold for the approximate Bahadur efficiencies of Kolmogorov-type tests.