Saddlepoint Methods and Statistical Inference

This paper reviews Daniels' saddlepoint approximation to the distribution of the mean of a random sample, and the many aspects of second order asymptotic inference that have been developed from it. These include Barndorff-Nielsen's approximation to the distribution of the maximum likelihood estimate, Bartlett factors for the likelihood ratio statistic and approximations to predictive and conditional likelihood. The emphasis is on statistical applications of the saddlepoint method. The intention is to provide fairly broad coverage of the literature and to indicate possibilities for future development. An annotated bibliography is included.