The Lepage Location-Scale Test Revisited

Generally, to make inferences about possible difference between two populations, a test for location is considered. Sometimes, there is more interest in scale differences rather than in location ones (e.g., industrial quality control). This paper is instead focused on joint nonparametric testing for location and scale. A test by Lepage [4] is considered. This is a rank test based on a combination of the Wilcoxon-Mann-Whitney test for location and the Ansari-Bradley test for scale. It is shown that the Lepage idea may be developed and extended within the nonparametric combination framework, for example by considering different combining functions or different test statistics. Moreover, it is easy to adopt a weighting testing strategy. The classical version of the Lepage test has been compared via simulation with various permutation versions of it, partly developed within the nonparametric combination framework. The rank test of Cucconi [1] for location-scale problem has been considered as well. It is shown that the tests maintain the type-one error rate close to the nominal level. The classical Lepage test works well and like the nonparametric combination test based on the Fisher combining function which is more flexible. A multivariate version of the classical Lepage test is presented.