The saddlepoint approximation for averages of conditionally independent random variables

Ziang Niu; Jyotishka Ray Choudhury; Eugene Katsevich

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arXiv · 2407.08915

The saddlepoint approximation for averages of conditionally independent random variables

Ziang Niu·Jyotishka Ray Choudhury·Eugene Katsevich

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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.

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Ziang Niu, Jyotishka Ray Choudhury, Eugene Katsevich. 2024-07-12. The saddlepoint approximation for averages of conditionally independent random variables. https://arxiv.org/abs/2407.08915

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