Motivated by the application of saddlepoint approximations to resampling-based statistical tests, we prove that the Lugannani-Rice formula has vanishing relative error when applied to approximate conditional tail probabilities of averages of conditionally independent random variables. In a departure from existing work, this result is valid under only sub-exponential assumptions on the summands, and does not require any assumptions on their smoothness or lattice structure. The derived saddlepoint approximation result can be directly applied to resampling-based hypothesis tests, including bootstrap, sign-flipping and conditional randomization tests. We exemplify this by providing the first rigorous justification of a saddlepoint approximation for the sign-flipping test of symmetry about the origin, initially proposed in 1955. On the way to our main result, we establish a conditional Berry-Esseen inequality for sums 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Lugannani-Rice公式在近似条件独立随机变量均值的条件尾部概率时具有可忽略的相对误差。与现有研究不同，该结果仅需对求和项作次指数假设，且不要求其光滑性或格点结构的任何假设。推导出的鞍点近似结果可直接应用于基于重采样的假设检验，包括自助法、符号翻转和条件随机化检验。我们通过首次严格论证1955年提出的原点对称性符号翻转检验的鞍点近似来例证这一应用。在证明主要结果的过程中，我们建立了条件独立随机变量和的条件Berry-Esseen不等式，该结论可能具有独立的研究价值。