Testing for central symmetry

•We propose two omnibus tests for bivariate central symmetry.•The two tests are asymptotically distribution-free.•The tests can be used for testing exchangeability after a data transformation.

Omnibus tests for central symmetry of a bivariate probability distribution are proposed. The test statistics compare empirical measures of opposite regions. Under rather weak conditions, we establish the asymptotic distribution of the test statistics under the null hypothesis; it follows that they are asymptotically distribution-free. After a simple transformation of the data, tests for central symmetry can also be employed for testing exchangeability. In a simulation study the good finite sample performance of the test procedures is shown.