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).
If the moment generating function of a distribution is written as
M(t)
K(t)=log(M(t))
\hat{f}(x)=
1 | |
\sqrt{2\piK''(\hat{s |
)}}\exp(K(\hat{s})-\hat{s}x)
\hat{F}(x)=\begin{cases}\Phi(\hat{w})+\phi(\hat{w})(
1 | |
\hat{w |
\hat{s}
K'(\hat{s})=x
\hat{w}=sgn{\hat{s}}\sqrt{2(\hat{s}x-K(\hat{s}))}
\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)
f(x)
f(x)
f(x)