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 $$M(t)$$ and the cumulant generating function as $$K(t) = \log(M(t))$$ then the saddlepoint approximation to the PDF of a distribution is defined as:
 * $$\hat{f}(x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) $$

and the saddlepoint approximation to the CDF is defined as:


 * $$\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} $$ where $$\hat{s}$$ is the solution to $$K'(\hat{s}) = x$$, $$\hat{w} = \sgn{\hat{s}}\sqrt{2(\hat{s}x - K(\hat{s}))}$$ and $$\hat{u} = \hat{s}\sqrt{K''(\hat{s})}$$.

When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function $$F(x)$$ may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function $$f(x)$$ (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 $$f(x)$$. Unlike the original saddlepoint approximation for $$f(x)$$, this alternative approximation in general does not need to be renormalized.