Note on the Consistency of Some Distribution-Free Tests for Dispersion

Abstract In this paper the consistency of the two-sample tests of Sukhatme [6], Ansari and Bradley [1], Siegel and Tukey [5], and Mood [3] is investigated. Each of these tests is a distribution-free analogue of the F-test for testing the equality of the variances of two normal distributions. If the two samples are taken from continuous distributions with the same median (say 0) and distribution functions F(x) and F(x/a) respectively, the hypothesis tested states that a = 1 and the two-sided tests are consistent for a ≠ 1. If, however, the samples are from continuous distributions with the same median 0 and arbitrary distribution functions F and G, little is known about their asymptotic properties. In this paper the conditions for consistency for this case are given; further it is investigated if the corresponding estimates of differences in dispersion satisfy the usual properties of measures of dispersion, namely 1) that X and − X have the same dispersion and 2) that, if X has a larger (or equal or smalle...