Saddlepoint approximations to the two-sample permutation tests

A saddlepoint approximation for a two-sample permutation test was obtained by Robinson^[7]. Although the approximation is very accurate, the formula is very complicated and difficult to apply. In this paper, we shall revisit the same problem from a different angle. We shall first turn the problem into a conditional probability and then apply a Lugannani-Rice type formula to it, which was developed by Skovagard^[8] for the mean of i.i.d. samples and by Jing and Robinson^[5] for smooth function of vector means. Both the Lugannani-Rice type formula and Robinson's formula achieve the same relative error of order O(n^-3/2), but the former is very compact and much easier to use in practice. Some numerical results will be presented to compare the two formulas