Edgeworth and Saddlepoint Approximations Compared

Why Compare the Methods?

Edgeworth and saddlepoint approximations both refine a normal approximation, but they do so in different ways. Edgeworth expansions add cumulant-based correction terms to a normal density or distribution function. Saddlepoint methods instead use the cumulant-generating function at a value chosen to match the point or threshold of interest. These differences can matter most away from the center, where skewness is pronounced, and for discrete variables, where the lattice structure cannot be ignored.

The comparison below uses a binomial point probability and upper tail, followed by the density and upper tail of a gamma distribution. The calculations illustrate a practical principle: neither method is uniformly superior in every setting. Accuracy depends on the target, the distribution, and whether the approximation respects the variable’s support and lattice structure.

Setup: For a random variable with cumulant-generating function $K(t)=\log E[e^{tX}]$, a saddlepoint for a target value $x$ solves $K'(t)=x$. Where the solution exists and $K''(t)>0$, the continuous saddlepoint density approximation is $$ \widehat f_{\mathrm{SP}}(x)=\frac{\exp\{K(t)-tx\}}{\sqrt{2\pi K''(t)}}. $$ An Edgeworth approximation instead uses a normal approximation plus terms involving standardized cumulants. The formulas below are first-order Edgeworth approximations; they are asymptotic approximations, not necessarily valid probability distributions at every finite sample size.

Worked Comparison: A Binomial Point Probability

Let $X\sim\operatorname{Binomial}(40,0.2)$, the number of successes in 40 independent trials with success probability 0.2. Its mean is $np=8$, its variance is $np(1-p)=6.4$, and its standardized skewness is $$ \gamma_1=\frac{1-2p}{\sqrt{np(1-p)}}=\frac{0.6}{\sqrt{6.4}}\approx 0.23717. $$ We compare the exact mass $P(X=12)$ with first-order local Edgeworth and lattice saddlepoint approximations.

Worked Example: The Binomial Mass at 12

Exact probability. Directly from the binomial formula, $$ P(X=12)=\binom{40}{12}(0.2)^{12}(0.8)^{28}\approx 0.0443. $$

First-order local Edgeworth approximation. The standardized target and lattice spacing on the standardized scale are $$ z=\frac{12-8}{\sqrt{6.4}}\approx1.58114, \qquad h=\frac{1}{\sqrt{6.4}}\approx0.39528. $$ For a lattice sum, the first-order local approximation is $h\phi(z)\{1+\gamma_1(z^3-3z)/6\}$, where $\phi$ is the standard normal density. Here $z^3-3z\approx-0.79057$, so the correction factor is $$ 1+\frac{0.23717}{6}(-0.79057)=0.96875. $$ Since $\phi(1.58114)\approx0.11418$, the approximated mass is $$ 0.39528(0.11418)(0.96875)\approx0.0437. $$

Lattice saddlepoint approximation. For the binomial, $$ K(t)=40\log(0.8+0.2e^t). $$ At $k=12$, the saddlepoint solves $K'(t)=12$. Equivalently, with $q=k/n=0.3$, $$ t=\log\left(\frac{q(1-p)}{p(1-q)}\right) =\log\left(\frac{12}{7}\right)\approx0.53900. $$ At this point $K''(t)=nq(1-q)=8.4$, and $K(t)-12t=-nD(q\Vert p)$, where $$ D(q\Vert p)=q\log(q/p)+(1-q)\log\{(1-q)/(1-p)\}\approx0.0281676. $$ The lattice saddlepoint mass approximation is $$ \frac{\exp\{-40D(0.3\Vert0.2)\}}{\sqrt{2\pi(8.4)}}\approx0.0446. $$

The exact mass is about 0.0443. In this case the saddlepoint value, about 0.0446, is closer than the first-order Edgeworth value, about 0.0437. Both are reasonably close, but this single point does not establish a general ranking.

Why the Binomial Saddlepoint Formula Has This Form

For this example, the lattice saddlepoint formula can also be understood through Stirling’s approximation. This gives a direct justification for the expression used above and explains its relationship to the exact binomial probability.

Derivation: For $k=nq$, where $0<k<n$, Stirling’s formula gives $m!=\sqrt{2\pi m}(m/e)^m\{1+O(1/m)\}$ as $m$ grows. Applying it to the binomial mass yields $$ \binom{n}{k}p^k(1-p)^{n-k} = \frac{\exp\{-nD(q\Vert p)\}}{\sqrt{2\pi n q(1-q)}}\{1+O(1/k+1/(n-k))\}. $$ Indeed, the exponential factor follows by collecting the powers of $n$, $k=nq$, and $n-k=n(1-q)$, while the square-root terms give $\sqrt{2\pi n q(1-q)}$ in the denominator. Since the saddlepoint has $K''(t)=nq(1-q)$ and $K(t)-tk=-nD(q\Vert p)$, the leading expression is exactly the lattice saddlepoint mass approximation. This derivation applies when $k$ and $n-k$ are sufficiently large for Stirling’s approximation to be useful; it is not a uniform relative-error guarantee at the boundaries. $\square$

Comparing Upper Tails

A point probability and a tail probability are different approximation targets. For a continuous distribution, the Lugannani–Rice formula is a common saddlepoint approximation to a tail. In the right-tail case, define $$ w=\sqrt{2\{tx-K(t)\}}, \qquad u=t\sqrt{K''(t)}, $$ where $K'(t)=x$ and $t>0$. The approximation is $$ P(X\ge x)\approx1-\Phi(w)+\phi(w)\left(\frac{1}{u}-\frac{1}{w}\right). $$ For a lattice variable, the corresponding formula must account for the lattice; in the binomial right-tail calculation below, the adjusted quantity is $u_{\mathrm{lat}}=(1-e^{-t})\sqrt{K''(t)}$. The lattice adjustment is not optional when approximating a discrete tail.

Worked Example: A Binomial Upper Tail

For the same $X\sim\operatorname{Binomial}(40,0.2)$, consider $P(X\ge12)$. Summing the binomial probabilities gives approximately $0.0875$. At $k=12$, use $t\approx0.53900$, $K''(t)=8.4$, and $2\{tk-K(t)\}=2nD(0.3\Vert0.2)\approx2.25340$. Thus $$ w\approx1.50113, \qquad u_{\mathrm{lat}}=(1-e^{-t})\sqrt{8.4} =\left(1-\frac{7}{12}\right)\sqrt{8.4} \approx1.2076. $$ The lattice Lugannani–Rice approximation is $$ 1-\Phi(1.50113)+\phi(1.50113) \left(\frac{1}{1.2076}-\frac{1}{1.50113}\right) \approx0.0876. $$ This tail approximation is close to the directly summed value in this example. Omitting the lattice adjustment would change the correction term and would not be the appropriate discrete-tail calculation.

Worked Comparison: A Skewed Continuous Density

Let $S$ be the sum of five independent exponential random variables, each with rate 1. Then $S$ has a gamma distribution with shape 5 and rate 1. Its cumulant-generating function is $K(t)=-5\log(1-t)$ for $t<1$; its mean is 5, variance is 5, and standardized skewness is $2/\sqrt5$. We compare the density at $x=8$, three units above the mean.

Worked Example: Density at 8 for a Gamma Sum

Exact density. The gamma density is $$ f_S(x)=\frac{e^{-x}x^4}{4!},\qquad x>0. $$ Therefore $$ f_S(8)=\frac{e^{-8}8^4}{4!}\approx0.05725. $$

First-order Edgeworth density. Here $z=(8-5)/\sqrt5=3/\sqrt5\approx1.34164$ and $\gamma_1=2/\sqrt5\approx0.89443$. The first-order standardized density correction is $1+\gamma_1(z^3-3z)/6$. Since $z^3-3z\approx-1.60997$, the correction factor is 0.76 (to the shown precision). Returning to the original scale divides the standardized density by $\sqrt5$, so $$ \frac{\phi(3/\sqrt5)}{\sqrt5}(0.76)\approx0.0551. $$

Saddlepoint density. Solving $K'(t)=5/(1-t)=8$ gives $t=3/8$, and $$ K''(t)=\frac{5}{(1-t)^2}=12.8, \qquad K(t)-8t=5\log(8/5)-3\approx-0.649982. $$ Thus $$ \widehat f_{\mathrm{SP}}(8)= \frac{\exp\{5\log(8/5)-3\}}{\sqrt{2\pi(12.8)}} \approx0.0582. $$ The exact density is about 0.05725: the saddlepoint approximation is slightly high, and the first-order Edgeworth approximation is slightly low. In absolute error, the saddlepoint value is closer for this particular density evaluation.

Worked Comparison: A Continuous Upper Tail

The same gamma model lets us compare tail approximations without changing distributions. Its exact survival probability has a finite-sum expression, while the two approximations use different information about the distribution.

Worked Example: The Gamma Survival Probability at 8

For a gamma variable with integer shape 5 and rate 1, integration by parts gives $$ P(S\ge8)=e^{-8}\sum_{j=0}^{4}\frac{8^j}{j!} =e^{-8}(1+8+32+85.3333+170.6667) \approx0.09963. $$

For the first-order Edgeworth tail, the standardized threshold is $z=3/\sqrt5$, and the CDF approximation is $$ P\left(\frac{S-5}{\sqrt5}\le z\right) \approx\Phi(z)+\frac{\gamma_1}{6}(1-z^2)\phi(z). $$ Because $z^2=1.8$, this gives a right-tail approximation $$ 1-\Phi(z)+\frac{\gamma_1}{6}(z^2-1)\phi(z)\approx0.1092. $$

For the Lugannani–Rice approximation, the saddlepoint remains $t=3/8$. Hence $w=\sqrt{2(8t-K(t))}\approx1.1402$ and $u=t\sqrt{K''(t)}\approx1.3416$. Substitution gives $$ 1-\Phi(w)+\phi(w)\left(\frac{1}{u}-\frac{1}{w}\right)\approx0.0997. $$ The exact tail is about 0.09963. Here the saddlepoint tail is much closer than the first-order Edgeworth tail. The example illustrates, but does not prove, a common advantage of saddlepoint tail formulas in skewed distributions.

Support, Lattices, and Computation

Edgeworth formulas are polynomial corrections to normal approximations. They do not automatically enforce the support of the original variable. For example, a density approximation built from a normal density is generally nonzero on negative values even when the exact gamma variable is positive. At sufficiently extreme standardized values, the polynomial correction can also make an approximation negative. A lattice Edgeworth approximation incorporates the spacing of a discrete distribution, but that adjustment alone does not force all approximated masses to be nonnegative or confine them to the finite binomial support $0,\ldots,n$.

Saddlepoint methods use the cumulant-generating function and its domain, which often reflects support more naturally. In the gamma example, $K(t)$ is defined for $t<1$, and a finite saddlepoint solving $K'(t)=x$ exists for $x>0$, matching the positive support. For the binomial, the saddlepoint mass calculation is for an interior value $0<k<n$; it should not be treated as a boundary formula for $k=0$ or $k=n$. Support awareness is valuable, but it does not make every saddlepoint approximation automatically accurate or valid at every boundary.

In computational terms, an Edgeworth approximation usually needs a normal density or CDF and a small number of cumulants. It is often inexpensive and convenient near the center, particularly when the standardized skewness and higher-order corrections are modest. A saddlepoint calculation requires evaluating $K(t)$, solving $K'(t)=x$, and evaluating derivatives such as $K''(t)$. That root may have a closed form, as in these examples, or require numerical work. The added calculation can be worthwhile for tail probabilities or pointwise accuracy, but it depends on a well-behaved cumulant-generating function and a reliable saddlepoint computation.

Common pitfall: It is wrong to conclude that an approximation is reliable because it matches the exact value at one central point, or to compare a continuous density approximation directly with a discrete probability mass. A density has units of inverse measurement, whereas a mass is a probability. For a lattice variable, use a local approximation with the lattice spacing for a point mass and a lattice-adjusted formula for a tail. Compare each method with the exact quantity of the same type and at the same target.

How to Choose Between Them

The examples suggest a disciplined comparison rather than a universal ranking. First identify whether the target is a density, point mass, CDF, or tail. Then use a formula appropriate to that target, including lattice adjustments when needed. Finally, check support and evaluate against an exact calculation or a trusted numerical method when one is available.

Edgeworth approximations expose how skewness and other cumulants perturb the normal approximation, which makes them informative as well as computationally light. Saddlepoint approximations adapt to the target through exponential tilting and can give strong numerical performance, especially for tails. Neither should be extrapolated beyond its conditions without diagnostics: a finite-order Edgeworth correction can behave poorly far into a tail, while saddlepoint calculations can be inaccurate when the root or the relevant cumulant function is problematic.

Check Your Understanding

Use the formulas and distinctions in this comparison to answer the following questions.

For $X\sim\operatorname{Binomial}(40,0.2)$, why is the lattice spacing included in the local Edgeworth approximation to $P(X=12)$?
In the binomial point-mass example, which approximation is closer to the exact mass, and what are the three values to the shown precision?
For the gamma sum, state the exact density at 8 and the first-order Edgeworth and saddlepoint density approximations.
Why does a continuous Edgeworth density approximation fail to respect the positive support of the gamma variable automatically?
Give one computational advantage of Edgeworth methods and one situation in which saddlepoint methods may justify their additional calculation.