Saddlepoint approximation method explained

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)=

	1
	\sqrt{2\piK''(\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})(

	1
	\hat{w

} - \frac) & \text x \neq \mu \\ \frac + \frac & \text x = \mu \end 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.