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).

Definition

If the moment generating function of a distribution is written as [math]\displaystyle{ M(t) }[/math] and the cumulant generating function as [math]\displaystyle{ K(t) = \log(M(t)) }[/math] then the saddlepoint approximation to the PDF of a distribution is defined as:

[math]\displaystyle{ \hat{f}(x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) }[/math]

and the saddlepoint approximation to the CDF is defined as:

[math]\displaystyle{ \hat{F}(x) = \begin{cases} \Phi(\hat{w}) + \phi(\hat{w})(\frac{1}{\hat{w}} - \frac{1}{\hat{u}}) & \text{for } x \neq \mu \\ \frac{1}{2} + \frac{K'''(0)}{6 \sqrt{2\pi} K''(0)^{3/2}} & \text{for } x = \mu \end{cases} }[/math]

where [math]\displaystyle{ \hat{s} }[/math] is the solution to [math]\displaystyle{ K'(\hat{s}) = x }[/math], [math]\displaystyle{ \hat{w} = \sgn{\hat{s}}\sqrt{2(\hat{s}x - K(\hat{s}))} }[/math] and [math]\displaystyle{ \hat{u} = \hat{s}\sqrt{K''(\hat{s})} }[/math].

When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function [math]\displaystyle{ F(x) }[/math] may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function [math]\displaystyle{ f(x) }[/math] (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 [math]\displaystyle{ f(x) }[/math]. Unlike the original saddlepoint approximation for [math]\displaystyle{ f(x) }[/math], 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 
• 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 
• 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