Title: The saddlepoint approximation for averages of conditionally independent random variables Show Chinese title

Title: 条件独立随机变量平均值的鞍点逼近

Abstract: 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. Show Chinese abstract

Abstract: 受将鞍点近似应用于基于重采样的统计检验的启发，我们证明了当用于近似条件独立随机变量平均值的条件尾部概率时，Lugannani-Rice公式具有消失的相对误差。 与现有工作不同，此结果仅在求和项上具有次指数假设，而不需要对其平滑性或格结构做出任何假设。 得出的鞍点近似结果可以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。 我们通过提供对关于原点对称性的符号翻转检验的鞍点近似的第一严格证明来说明这一点，该检验最初在1955年提出。 在得到主要结果的过程中，我们建立了条件独立随机变量之和的条件Berry-Esseen不等式，这可能具有独立兴趣。