This paper derives higher-order terms in the double-saddlepoint expansion of Skovgaard for a unidimensional conditional cumulative distribution function. Expansions for continuous and lattice random variables are derived. Results are applied to the sufficient statistic in logistic regression.
"Higher-order approximations to conditional distribution functions." Ann. Statist. 24 (1) 353 - 364, February 1996. https://doi.org/10.1214/aos/1033066213