Saddlepoint p-values for two-sample bivariate tests

A double saddlepoint approximation for the exact p-values of a class of two-sample bivariate tests is derived. The saddlepoint method approximates the underlying permutation distributions of the tests. Some widely used bivariate tests are shown to be in this class. Extensive simulation studies are used to evaluate the performance of the saddlepoint approximation. The saddlepoint approximation is compared with the asymptotic approximation and shows great improvement in accuracy over the asymptotic normal approximation.