Saddlepoint approximation for the p-values of some distribution-free tests

This article discusses the saddlepoint approximation for the p-values of some distribution-free tests, a signed rank test for bivariate location problems and a dispersion test for scale problems. The statistics of the two considered tests are constructed based on the ratio of two variables. The accuracy of the saddlepoint approximation is compared to traditional asymptotic normal approximation by applying numerical comparisons. Furthermore, the proposed approximations are illustrated by analyzing numerical examples. The results of numerical comparisons indicate that the approximation error resulting from the proposed method is much lower than the traditional method, which is evidence of the superiority of the proposed approximation method over the traditional method. Accordingly, we can say that the saddlepoint approximation method can be a competitive alternative to the traditional method.