Saddlepoint approximation of the p-values for the multivariate one-sample sign and signed-rank tests

A multivariate data analysis (MVDA) is a powerful statistical approach to simultaneously analyze datasets with multiple variables. Unlike univariate or bivariate analyses, which simultaneously focus on one or two variables, respectively, MVDA considers the interactions and relationships among multiple variables within a dataset. Several nonparametric tests can be used in the context of one-sample multivariate location problems. The exact distributions of such tests cannot be analytically computed and are usually approximated using an asymptotic approximation. This article proposes the saddlepoint approximation method to approximate the tail probability for multivariate sign and signed-rank tests. It is suggested as a more accurate alternative to the traditional asymptotic approximation method and an alternative to the simulation method. It requires a lot of time as it depends on all possible permutations. Real data examples were provided to illustrate the calculation of p-values, and a simulation study was conducted to compare the accuracy of the saddlepoint approximation method with the simulation method (permutation-based, so time-consuming) and an asymptotic normal approximation method. The study results show that the saddlepoint approximation provides highly accurate approximations to the p-values of the considered statistics, and it often outperforms the normal approximation. Additionally, the results show that the proposed method's computation time is much less than that of the time-consuming simulation method.