  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.
