Review about the Permutation Approach in Hypothesis Testing

Today, permutation tests represent a powerful and increasingly widespread tool of statistical inference for hypothesis-testing problems. To the best of our knowledge, a review of the application of permutation tests for complex data in practical data analysis for hypothesis testing is missing. In particular, it is essential to review the application of permutation tests in two-sample or multi-sample problems and in regression analysis. The aim of this paper is to consider the main scientific contributions on the subject of permutation methods for hypothesis testing in the mentioned fields. Notes on their use to address the problem of missing data and, in particular, right-censored data, will also be included. This review also tries to highlight the limits and advantages of the works cited with a critical eye and also to provide practical indications to researchers and practitioners who need to identify flexible and distribution-free solutions for the most disparate hypothesis-testing problems.