Saddlepoint approximation method

The saddlepoint approximation method, initially proposed by Daniels (1954) 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 (1980). The saddlepoint approximation method, initially proposed by Daniels (1954) 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 (1980). If the moment generating function of a distribution is written as M ( t ) {displaystyle M(t)} and the cumulant generating function as K ( t ) = log ⁡ ( M ( t ) ) {displaystyle K(t)=log(M(t))} then the saddlepoint approximation to the PDF of a distribution is defined as: