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 ${\displaystyle N}$ 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 ${\displaystyle X=\sum _{i=1}^{N}X_{i}}$ 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 ${\displaystyle K(t)=\log(M(t))=\sum _{i=1}^{N}\log E\left[e^{tX_{i}}\right]}$ then the saddlepoint approximation to the PDF of the distribution ${\displaystyle X}$ is defined as:^[1]

${\displaystyle {\hat {f}}_{X}(x)={\frac {1}{\sqrt {2\pi K''({\hat {s}})}}}\exp(K({\hat {s}})-{\hat {s}}x)\,\left(1+{\mathcal {R}}\right)}$

where ${\displaystyle {\mathcal {R}}}$ contains higher order terms to refine the approximation^[1] and the saddlepoint approximation to the CDF is defined as:^[1]

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

where ${\displaystyle {\hat {s}}}$ is the solution to ${\displaystyle K'({\hat {s}})=x}$, ${\displaystyle {\hat {w}}=\operatorname {sgn} {\hat {s}}{\sqrt {2({\hat {s}}x-K({\hat {s}}))}}}$ ,${\displaystyle {\hat {u}}={\hat {s}}{\sqrt {K''({\hat {s}})}}}$, and ${\displaystyle \Phi (t)}$ and ${\displaystyle \phi (t)}$ are the cumulative distribution function and the probability density function of a normal distribution, respectively, and ${\displaystyle \mu }$ is the mean of the random variable ${\displaystyle X}$:

${\displaystyle \mu \triangleq E\left[X\right]=\int _{-\infty }^{+\infty }xf_{X}(x)\,{\text{d}}x=\sum _{i=1}^{N}E\left[X_{i}\right]=\sum _{i=1}^{N}\int _{-\infty }^{+\infty }x_{i}f_{X_{i}}(x_{i})\,{\text{d}}x_{i}}$.

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

• 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, JSTOR 2335316
• 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, JSTOR 1426607, S2CID 124484743
• 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