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).
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 .
When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function (Routledge and Tsao, 1997). This result establishes the derivative of a truncated Lugannani and Rice series as an alternative asymptotic approximation for the density function . Unlike the original saddlepoint approximation for , this alternative approximation in general does not need to be renormalized.
References
- Butler, Ronald W. (2007), Saddlepoint approximations with applications, Cambridge: Cambridge University Press, ISBN 9780521872508
- Daniels, H. E. (1954), "Saddlepoint Approximations in Statistics", The Annals of Mathematical Statistics, 25 (4): 631–650, doi:10.1214/aoms/1177728652
- Daniels, H. E. (1980), "Exact Saddlepoint Approximations", Biometrika, 67 (1): 59–63, doi:10.1093/biomet/67.1.59, JSTOR 2335316
- Lugannani, R.; Rice, S. (1980), "Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables", Advances in Applied Probability, 12 (2): 475–490, doi:10.2307/1426607, JSTOR 1426607, S2CID 124484743
- Reid, N. (1988), "Saddlepoint Methods and Statistical Inference", Statistical Science, 3 (2): 213–227, doi:10.1214/ss/1177012906
- Routledge, R. D.; Tsao, M. (1997), "On the relationship between two asymptotic expansions for the distribution of sample mean and its applications", Annals of Statistics, 25 (5): 2200–2209, doi:10.1214/aos/1069362394