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Bounds on AREs for restricted classes of distributions defined via tail- orderings. (English) Zbl 0598.62051

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 W. R. van Zwet [Convex transformations of random variables. Math. Centre, Tracts 7 (1964; Zbl 0125.371)], M. J. Lawrence [Ann. Stat. 3, 413-428 (1975; Zbl 0305.62029)], R. E. Barlow and F. Proschan [Statistical theory of reliability and life testing. (1975; Zbl 0379.62080)], 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.

MSC:

62G20 Asymptotic properties of nonparametric inference
62E10 Characterization and structure theory of statistical distributions
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