Invariant tests for symmetry about an unspecified point based on the empirical characteristic function

This paper considers a flexible class of omnibus affine invariant tests for the hypothesis that a multivariate distribution is symmetric about an unspecified point. The test statistics are weighted integrals involving the imaginary part of the empirical characteristic function of suitably standardized given data, and they have an alternative representation in terms of an L$^2$-distance of nonparametric kernel density estimators. Moreover, there is a connection with two measures of multivariate skewness. The tests are performed via a permutational procedure that conditions on the data.