A test for symmetries of multivariate probability distributions

A Monte Carlo test for multivariate symmetries is proposed. The Monte Carlo simulations are performed conditionally on a minimal sufficient statistic for the class of distributions with symmetric density. Additionally, a general purpose test statistic based on a distance measure between the probability density function and its symmetrised version is introduced. The Monte Carlo tests for spherical symmetry and multivariate reflection symmetry are studied numerically for this statistic and the results indicate that the tests perform well compared to other tests. The method is illustrated with an analysis of a real dataset.