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 smaller) dispersion than Y, − X has a larger (or equal or smaller) dispersion than Y .