Conditional saddlepoint approximations for non-continuous and non-lattice distributions

This manuscript presents an approximation to the distribution function of a smooth transformation of a random vector, conditional on the event that values of other smooth transformations of the same random vector lie in a small rectangle. This approximation is used to extend the application of standard saddlepoint conditional tail area approximations in circumstances beyond continuous and lattice cases currently justified in the literature. We consider application to two examples, finite sampling and score testing in logistic regression, where conditioning on a rectangle is essential.