Testing symmetry about an unknown median, via linear rank procedures

Abstract

The properties of several linear rank statistics for testing the hypothesis that a random variable with unknown median has a symmetric distribution, are explored. Each test is related, in a natural way, to a rank type estimator of the median. Specifically, the tests are based on swre functions which are the third element of a complete orthonormal basis for L ₂[O,1] and the median is estimated by a natural associated R-estimator. It is shown that by linking the score function and estimator in this fashion the resulting tests have a simple asymptotic distribution under both the null hypothesis and sequences of local alternatives. Based on the asymptotic distribution theory, simple small sample modifications are presented. It is shown that for small n, even when approximate asymptotic critical values are used, these test maintain a stable a-level, which is close to the nominal level, over a wide range of symmetric distributions. Also with respect to power, the tests are competitive with existing testing procedures.

Keywords:

• Tests of symmetry
• linear rank statistics