Motivated by the application of saddlepoint approximations to resampling-based statistical tests, we prove that a Lugananni-Rice style approximation for conditional tail probabilities of averages of conditionally independent random variables has vanishing relative error. We also provide a general condition on the existence and uniqueness of the solution to the corresponding saddlepoint equation. The results are valid under a broad class of distributions involving no restrictions on the smoothness of the distribution function. The derived saddlepoint approximation formula can be directly applied to resampling-based hypothesis tests, including bootstrap, sign-flipping and conditional randomization tests. Our results extend and connect several classical saddlepoint approximation results. On the way to proving our main results, we prove a new conditional Berry-Esseen inequality for the sum of conditionally independent random variables, which may be of independent interest.

翻译：受鞍点近似在基于重采样的统计检验中应用的启发，我们证明了对于条件独立随机变量均值的条件尾部概率，Lugananni-Rice 型近似具有可忽略的相对误差。我们还给出了相应鞍点方程解存在唯一性的一般条件。该结果适用于广泛的分布类别，且无需对分布函数的光滑性施加任何限制。推导出的鞍点近似公式可直接应用于基于重采样的假设检验，包括自助法、符号翻转检验和条件随机化检验。我们的结果扩展并连接了若干经典的鞍点近似结论。在证明主要结果的过程中，我们为条件独立随机变量和证明了一个新的条件 Berry-Esseen 不等式，该不等式本身可能具有独立的研究价值。