Saddlepoint approximation for the bivariate two-sample tests under the random allocation design

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.