Saddlepoint approximation method


The saddlepoint approximation method, initially proposed by Daniels is a specific example of the mathematical saddlepoint technique applied to statistics. It provides a highly accurate approximation formula for any PDF or probability mass function of a distribution, based on the moment generating function. There is also a formula for the CDF of the distribution, proposed by Lugannani and Rice.

Definition

If the moment generating function of a distribution is written as and the cumulant generating function as then the saddlepoint approximation to the PDF of a distribution is defined as:
and the saddlepoint approximation to the CDF is defined as:
where is the solution to, and