# n'th cumulant (of a CGF) for exponential family / exponential dispersion model

The n'th cumulant is defined to be the n'th derivative of the CGF (cumulant generating function).

$$\kappa_n = \frac{d^n K(t)}{dt^n} |_{t=0}$$

But I'm reading in a book (p.215, chapter5, eq. 5.8) now that for the exponential family / exponential dispersion model, this is actually equal to:

$$K \underset{exp.}{=} \frac{\kappa(\theta+t\phi) - \kappa(\theta)}{\phi}\\ \kappa_n = \phi^{n-1} \frac{d^n\kappa(\theta)}{d\theta^n}$$

Where:

1. $$\theta$$ is the canonical/natural parameter in exponential family.

2. $$K$$ is cumulant generating function

3. $$\kappa_n$$ is the nth cumulant

I'm not really sure how come you get this result. The $$\phi^{n-1}$$ I understand, but not how the derivative changes from $$t$$ to $$\theta$$.

As you did'nt define your terms, I will use the definition of exponential dispersion model used in Definition of exponential family with dispersion parameter, which is $$f(y|\theta,\phi) = \exp\left(\frac{y\theta - b(\theta)}{a(\phi)} + c(y, \phi)\right)$$ which is an exponential family in $$\theta$$, for each fixed $$\phi$$. It seems you have in your notation $$a(\phi)=\phi$$. Then we can calculate the mgf (moment generating function) as $$M(t)=\exp\left( \frac{b(t a(\phi)+\theta)-b(\theta)}{a(\phi)} \right)$$ so the cumulant generating function $$K(t)=\log M(t)= \frac{b(t a(\phi)+\theta)-b(\theta)}{a(\phi)} .$$ Then $$K'(t)=\frac{b'(t a(\phi)+\theta)\cdot a(\phi)}{a(\phi)}=b'(t a(\phi)+\theta)$$ where $$'$$ means the derivative of a function with respect to its argument. It does not matter if that argument is named $$t$$ or $$\theta$$. Now it is easy to see that we can continue differentiating to get $$K^{(n)}(t)=b^{(n)}(t a(\phi)+\theta)\cdot a^{n-1}(\phi)$$ by repeated application of the chain rule. Now, if you set $$t=0$$ and revert to the $$\frac{d}{d t}$$ notation, you get the result. Now, since on the RHS the argument of $$b$$ is $$\theta$$, we differentiate there $$b$$ with respect to its argument, which explains your confusion.