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An Affine‐Invariant Generalization of the Wilcoxon Signed‐Rank Test for the Bivariate Location Problem
Author(s) -
Larocque Denis,
Tardif Serge,
Eeden Constance van
Publication year - 2003
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00271
Subject(s) - mathematics , bivariate analysis , wilcoxon signed rank test , sign test , test statistic , null distribution , invariant (physics) , univariate , statistics , null hypothesis , generalization , affine transformation , chi square test
This paper proposes an affine‐invariant test extending the univariate Wilcoxon signed‐rank test to the bivariate location problem. It gives two versions of the null distribution of the test statistic. The first version leads to a conditionally distribution‐free test which can be used with any sample size. The second version can be used for larger sample sizes and has a limiting χ 2 2 distribution under the null hypothesis. The paper investigates the relationship with a test proposed by Jan & Randles (1994). It shows that the Pitman efficiency of this test relative to the new test is equal to 1 for elliptical distributions but that the two tests are not necessarily equivalent for non‐elliptical distributions. These facts are also demonstrated empirically in a simulation study. The new test has the advantage of not requiring the assumption of elliptical symmetry which is needed to perform the asymptotic version of the Jan and Randles test.