Saddlepoint Approximations for Nonidentical Sums

Why Aggregate Saddlepoints Matter

A sum does not need identically distributed components to have a useful saddlepoint approximation. Independence is enough to combine the components’ cumulant-generating functions (CGFs). The resulting aggregate CGF describes the sum directly, and its saddlepoint is chosen to match the requested value of that sum.

This is useful when a total combines variables with different means, variances, or distributional shapes. Rather than force a common distribution onto the summands, retain their separate CGFs and add them. The standard saddlepoint density and tail formulas then use the aggregate CGF, its first two derivatives, and the solution of one scalar equation.

Definition: Let $X_1,\ldots,X_n$ be independent random variables, and let $S=\sum_{i=1}^n X_i$. For each $i$, define the CGF $K_i(t)=\log E[e^{tX_i}]$ where this expectation is finite. The aggregate CGF is $K(t)=\log E[e^{tS}]$. For a target value $s$, an aggregate saddlepoint is a value $\hat t$ in the interior of the domain of $K$ satisfying $K'(\hat t)=s$.

Combining CGFs and Finding the Root

Independence makes the aggregate calculation simple. If the individual CGFs are finite on a common open interval $I$, then, for $t\in I$,

E[e^{tS}] =E\!\left[\prod_{i=1}^n e^{tX_i}\right] =\prod_{i=1}^n E[e^{tX_i}], \qquad K(t)=\sum_{i=1}^n K_i(t).

Consequently, $K'(t)=\sum_i K_i'(t)$ and $K''(t)=\sum_i K_i''(t)$, whenever the derivatives exist. The saddlepoint equation is therefore

\sum_{i=1}^n K_i'(\hat t)=s.

It is an equation for the sum, not a separate saddlepoint equation for each summand. The target $s$ is usually a value of the total itself. If the target is instead a value $x$ of the sample average $S/n$, the corresponding total target in this equation is $s=nx$.

Theorem: Suppose $K$ is finite and twice differentiable on an open interval $I$, with $K''(t)>0$ for every $t\in I$. Then $K'$ is strictly increasing on $I$. For each $s$ in the range $K'(I)$, the equation $K'(t)=s$ has exactly one solution in $I$.

Proof. For any $a,b\in I$ with $a<b$, the mean value theorem gives a $c\in(a,b)$ such that

$$ K'(b)-K'(a)=K''(c)(b-a)>0. $$

Thus $K'$ is strictly increasing and can take the value $s$ at most once. If $s\in K'(I)$, it takes that value at least once by the definition of the range. The solution therefore exists and is unique. For an aggregate CGF, $K''(t)$ is the variance of $S$ under exponential tilting; it is positive whenever the tilted sum is nondegenerate. $\square$

Density Approximation from Exponential Tilting

Assume the summands have CGFs finite near $\hat t$, where $K'(\hat t)=s$, and that $S$ has a sufficiently regular continuous density. Exponential tilting changes the distribution so that $s$ is the mean of the tilted sum. The density of the original sum can be recovered exactly from the tilted density; the saddlepoint approximation then uses a local Gaussian approximation at that tilted mean.

Density approximation: Under conditions that justify a local Gaussian approximation to the tilted density at $s$, the saddlepoint approximation is $$ \widehat f_S(s)= \frac{\exp\{K(\hat t)-\hat t s\}} {\sqrt{2\pi K''(\hat t)}}, \qquad K'(\hat t)=s. $$

Here is the exact change-of-measure identity behind the formula. Define a tilted joint distribution by multiplying the original joint density (or probability law) by $e^{tS-K(t)}$. This multiplier integrates to one because $E[e^{tS}]=e^{K(t)}$. Write $f_{S,t}$ for the density of $S$ under this tilted distribution. The relation between the original and tilted densities is

f_S(x)=e^{K(t)-tx}f_{S,t}(x).

At $t=\hat t$, differentiation of the CGF gives $E_{\hat t}[S]=K'(\hat t)=s$ and $\operatorname{Var}_{\hat t}(S)=K''(\hat t)$. If the tilted density is locally approximated at its mean by a normal density with this variance, then $f_{S,\hat t}(s)$ is approximated by $1/\sqrt{2\pi K''(\hat t)}$. Substitution into the exact identity gives the displayed saddlepoint density. The change-of-measure identity is exact; the local Gaussian step is the approximation, and its accuracy depends on the distribution and asymptotic regime.

Worked Example: Two Exponentials with Different Rates

Let $X_1$ and $X_2$ be independent exponential variables with rates 1 and 2, respectively. Their CGFs are $K_1(t)=-\log(1-t)$ for $t<1$ and $K_2(t)=-\log(1-t/2)$ for $t<2$. For $S=X_1+X_2$, calculate the saddlepoint density approximation at $s=2$.

The common CGF domain is $t<1$, and the aggregate CGF and its first two derivatives are

K(t)=-\log(1-t)-\log(1-t/2),\qquad K'(t)=\frac{1}{1-t}+\frac{1}{2-t},\qquad K''(t)=\frac{1}{(1-t)^2}+\frac{1}{(2-t)^2}.

Setting $K'(t)=2$ gives

\frac{1}{1-t}+\frac{1}{2-t}=2 \quad\Longrightarrow\quad \frac{3-2t}{(1-t)(2-t)}=2 \quad\Longrightarrow\quad 2t^2-4t+1=0.

The roots are $t=1\pm1/\sqrt{2}$, and only $\hat t=1-1/\sqrt{2}\approx0.292893$ lies in the common domain $t<1$. The correct equation is

\frac{3-2t}{(1-t)(2-t)}=2 \quad\Longrightarrow\quad 2t^2-4t+1=0.

The root in the domain is $\hat t=1-1/\sqrt{2}\approx0.292893$. At this value, $1-\hat t=1/\sqrt2$, $2-\hat t=1+1/\sqrt2$, and $K''(\hat t)=2+1/(1+1/\sqrt2)^2\approx2.34315$. Substitution into the density formula gives the approximation evaluated below.

Worked Example: Correct Saddlepoint Calculation at a Different Target

For the same independent exponential variables with rates 1 and 2, take the target $s=2$. Since $K'(t)=1/(1-t)+1/(2-t)$, setting this equal to 2 gives $3-2t=2(1-t)(2-t)$, or $2t^2-4t+1=0$. Its in-domain root is $\hat t=1-1/\sqrt2\approx0.292893$, which satisfies $K'(\hat t)=2$.

At this root, $1-\hat t=1/\sqrt2\approx0.707107$ and $1-\hat t/2=(1+1/\sqrt2)/2\approx0.853553$. Therefore

K(\hat t)=-\log\!\left(\frac{1+\sqrt2}{4}\right) =\log\!\left(\frac{4}{1+\sqrt2}\right)\approx0.504921.

Also $2-\hat t=1+1/\sqrt2$, so

K''(\hat t)=2+\frac{1}{(1+1/\sqrt2)^2}\approx2.34315.

The exponent is $K(\hat t)-2\hat t\approx0.504921-0.585786=-0.080866$, and the denominator is $\sqrt{2\pi(2.34315)}\approx3.837$. Hence

\widehat f_S(2)\approx\frac{e^{-0.080866}}{3.837}\approx0.2404.

The calculation illustrates the central safeguard: check the saddlepoint by substituting it into $K'(\hat t)=s$ before evaluating the density formula.

Tail Approximation for an Aggregate Sum

For a continuous nonlattice sum, the Lugannani–Rice approximation uses the same aggregate saddlepoint. For $s\ne K'(0)$, define the signed deviance root $w$ and the scale $u$ by

w=\operatorname{sign}(\hat t) \sqrt{2\{\hat t s-K(\hat t)\}}, \qquad u=\hat t\sqrt{K''(\hat t)}.

Then the upper-tail approximation is

P(S\ge s)\ \approx\ 1-\Phi(w)+\phi(w)\left(\frac{1}{u}-\frac{1}{w}\right),

where $\Phi$ and $\phi$ are the standard normal distribution function and density. This formula is used away from the mean; at the mean the displayed terms are singular and the appropriate limiting form must be used. For a lattice sum, the continuous-tail formula is not automatically appropriate: lattice corrections are needed.

Worked Example: Tail Approximation for Heterogeneous Gamma Summands

Let $X_1\sim\operatorname{Gamma}(2,\text{ rate }1)$ and $X_2\sim\operatorname{Gamma}(1,\text{ rate }2)$, independently. Their CGFs are $-2\log(1-t)$ and $-\log(1-t/2)$, so $K(t)=-2\log(1-t)-\log(1-t/2)$. Approximate $P(S\ge14/3)$.

The derivatives are $K'(t)=2/(1-t)+1/(2-t)$ and $K''(t)=2/(1-t)^2+1/(2-t)^2$. At $\hat t=1/2$,

K'(1/2)=4+\frac{2}{3}=\frac{14}{3}, \qquad K''(1/2)=8+\frac{4}{9}=\frac{76}{9}.

Thus $K(1/2)=\log(16/3)\approx1.67398$, $\hat t s=7/3$, $w=\sqrt{2(7/3-\log(16/3))}\approx1.1484$, and $u=(1/2)\sqrt{76/9}\approx1.4530$. Using $1-\Phi(w)\approx0.1254$, $\phi(w)\approx0.2064$, and $1/u-1/w\approx-0.1826$, gives

P(S\ge14/3)\ \approx\ 0.1254+(0.2064)(-0.1826)\approx0.0877.

Both summands contribute to the aggregate CGF, but their shapes and rates remain distinct. No identically distributed assumption enters the calculation.

Normal Summands as a Check

A normal sum provides a useful check because the saddlepoint density approximation is exact in this case. If independent $X_i\sim N(\mu_i,\sigma_i^2)$, then

K(t)=t\sum_i\mu_i+\frac{t^2}{2}\sum_i\sigma_i^2.

Worked Example: Exactness for Two Unequal Normal Summands

Let $X_1\sim N(0,1)$ and $X_2\sim N(1,4)$ independently. Then $K(t)=t+2.5t^2$, $K'(t)=1+5t$, and $K''(t)=5$. At $s=3$, the saddlepoint is $\hat t=(3-1)/5=0.4$, and $K(\hat t)=0.8$. Therefore

\widehat f_S(3)= \frac{e^{0.8-0.4(3)}}{\sqrt{2\pi(5)}} =\frac{e^{-0.4}}{\sqrt{10\pi}} \approx0.1196.

The exact sum is $N(1,5)$, whose density at 3 is $\exp\{-(3-1)^2/(2\cdot5)\}/\sqrt{10\pi}$, the same value. Exactness here follows from normality; for general nonidentical summands the formula is an approximation.

Common Pitfalls and Practical Interpretation

Using a component’s saddlepoint. The root must solve $K'(\hat t)=s$ for the aggregate $K=\sum_i K_i$. Solving a separate equation for one $K_i$ does not generally center the tilted sum at $s$.
Ignoring the common domain. A numerical root outside the interval where every required CGF is finite is not a valid saddlepoint.
Confusing total and average targets. For $S/n=x$, solve $K'(\hat t)=nx$, not $K'(\hat t)=x$.
Treating the formula as exact. The density formula uses a local Gaussian approximation under the tilted law. It can be highly accurate, but it is not an identity except in special cases such as a normal sum.
Skipping arithmetic checks. Verify the root in $K'(\hat t)=s$, then independently evaluate $K(\hat t)$, $K''(\hat t)$, and the exponent $K(\hat t)-\hat t s$.

The main advantage for nonidentical sums is that heterogeneity is retained without changing the structure of the method: add the CGFs, solve the aggregate equation, and evaluate the density or tail formula using aggregate derivatives. Independence supplies the addition rule; suitable smoothness and local approximation conditions determine whether the resulting numerical approximation is reliable.

Check Your Understanding

Try these questions using the aggregate-CGF viewpoint.

For independent summands with CGFs $K_1,\ldots,K_n$, write the CGF and saddlepoint equation for their total $S$.
Why does $K''(t)>0$ imply that there is at most one saddlepoint for a given target within the CGF domain?
If the target is $x$ for the average $S/n$, what total target belongs in the saddlepoint equation?
Which part of the tilted-density argument is exact, and which part introduces the density approximation?
Why should a lattice sum not automatically be treated with the continuous Lugannani–Rice formula?