The saddlepoint approximation for averages of conditionally independent random variables

TL;DR

This paper proves that the Lugannani-Rice saddlepoint approximation achieves vanishing relative error for conditional tail probabilities of averages of conditionally independent variables under minimal assumptions, with applications to resampling tests.

Contribution

It establishes a new saddlepoint approximation result valid under sub-exponential conditions without smoothness or lattice assumptions, and applies it to justify resampling-based hypothesis tests.

Findings

Vanishing relative error of Lugannani-Rice formula for conditional tail probabilities.

First rigorous saddlepoint approximation for sign-flipping test of symmetry.

Conditional Berry-Esseen inequality for sums of conditionally independent variables.

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.