We introduce a nonparametric test statistic for the permutation test in complete block designs. We find the region in which the statistic exists and consider particularly its properties on the boundary of the region. Further, we prove that saddlepoint approximations for tail probabilities can be obtained inside the interior of this region. Finally, numerical examples are given showing that both accuracy and power of the new statistic improves on these properties of the classical $F$-statistic under some non-Gaussian models and equals them for the Gaussian case.
"A new permutation test statistic for complete block designs." Ann. Statist. 43 (1) 90 - 101, February 2015. https://doi.org/10.1214/14-AOS1266