Current status and panel count data appear in many applied fields, including medicine, clinical trials, epidemiology, econometrics, demography, engineering and public health. Therefore, in this article, we use the saddlepoint approximation method to approximate the exact p-value of a number of nonparametric tests for the current status and panel count data under a generalized permuted block design. The saddlepoint approximation is referred to as higher-order approximation and it is more accurate than the methods that lead to approximations that are accurate to the first order, such as the asymptotic normal approximation method. To verify the accuracy and efficiency of the saddlepoint approximation method, a simulation study is conducted. The simulation study results confirm that the saddlepoint approximation method is more powerful than the existing approximation method. Furthermore, number of real current status and panel count data sets are analyzed and displayed as illustrative examples.

This article addresses the challenge of accurately estimating p-values in bivariate two-sample tests under random allocation designs, a common setting in clinical and reliability studies. Existing normal approximations often perform poorly in small samples and in the distribution tails, leading to unreliable inference. To overcome this limitation, we propose the use of the saddlepoint approximation as a highly accurate alternative. Through simulation studies and real-data analyses, we demonstrate that the proposed method consistently yields p-values that are closer to the exact permutation values than those obtained from traditional normal approximations, particularly in small-sample settings.

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.

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.