Abstract
The article shows how some common measures of association between two random vectors may be used to test multivariate symmetry around a subspace (possibly up to a shift), which also permits testing exchangeability, axial symmetry, halfspace symmetry, and certain goodness-of-fit and equality-of-scale hypotheses. The resulting (parametric, nonparametric, permutation, and asymptotic) tests of the symmetry, consistent in the class of all elliptical distributions, are also illustrated with a few simulation and real data examples.