Nonparametric Tests for Scale

Abstract

This paper is concerned with two sample rank tests for scale alternatives. The two samples are assumed to have continuous distribution functions with the difference in respective location parameters (medians) known. Various rank tests are considered and compared from the point of view of limiting Pitman efficiency for normal and nonnormal alternatives. Among the tests considered is a test with efficiency one relative to the $F$-test for normal alternatives. Tables are given to facilitate its use. Small sample power and efficiency for normal alternatives are computed for equal sample sizes of 5. The small sample efficiency values differ appreciably from the limiting value; this deficiency of power appears to derive from the use of ranks per se rather than from the use of a rank test that is not optimal among rank tests. Lastly, a rank test is proposed for particular alternatives which is most powerful for rectangular densities. It is a simple test which is seen to have surprisingly good power for normal alternatives.