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:
:$\hat(x) = \frac \exp(K(\hat) - \hatx)$
and the saddlepoint approximation to the CDF is defined as:

:$\hat(x) = \begin \Phi(\hat) + \phi(\hat)(\frac - \frac) & \text x \neq \mu \\ \frac + \frac & \text x = \mu \end$
where $\hat$ is the solution to $K'(\hat) = x$, $\hat = \sgn\sqrt$ and $\hat = \hat\sqrt$.

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.

References

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* {{citation , last1=Routledge, first1=R. D. , last2=Tsao , first2=M. , title= On the relationship between two asymptotic expansions for the distribution of sample mean and its applications , journal=Annals of Statistics , volume=25 , issue=5 , pages=2200–2209 , year=1997 , doi=10.1214/aos/1069362394 , doi-access=free

Asymptotic analysis
Perturbation theory