The saddlepoint approximation method, initially proposed by Daniels (1954)[1] is a specific example of the mathematical saddlepoint technique applied to statistics, in particular to the distribution of the sum of
independent random variables.
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).
[2] If the moment generating function of a random variable
is written as
{\displaystyle M(t)=E\left[e^{tX}\right]=E\left[e^{t\sum _{i=1}^{N}X_{i}}\right]}
and the cumulant generating function as
( t ) = log (
then the saddlepoint approximation to the PDF of the distribution
is defined as:[1] where
contains higher order terms to refine the approximation[1] and the saddlepoint approximation to the CDF is defined as:[1] where
is the solution to
= sgn
{\displaystyle {\hat {w}}=\operatorname {sgn} {\hat {s}}{\sqrt {2({\hat {s}}x-K({\hat {s}}))}}}
ϕ ( t )
are the cumulative distribution function and the probability density function of a normal distribution, respectively, and
μ
is the mean of the random variable
μ ≜
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.