Testing for symmetry in multivariate distributions

Using the empirical characteristic function, a Cramér–von Mises test for reflected symmetry about an unspecified point is derived for multivariate distributions. The test statistic is based on an empirical process for which the weak convergence is established. The null properties of the test are studied as well as its power and local power. Estimators for the unknown symmetric point are previously proposed. Their consistency and asymptotical normality are proved by studying the weak convergence of some multidimensional empirical process. A simulation experiment shows that the estimators of the symmetric point are good, and that the test performs well on the examples tested. The new test is compared to the one derived in [N. Henze, B. Klar, S.G. Meintanis, Invariant tests for symmetry about an unspecified point based on empirical characteristic function, J. Multivariate. Anal. 87 (2003) 275–297].