# Exponential Family with Dispersion Parameter Distributions

Defining the exponential dispersion family by

$exp\left(\frac{x_{i}\theta - b(\theta_{i})}{\phi} + c(x_{i}, \phi, w_{i})\right)$

I'd like to change the usual Inverse-Gaussian density below to the form above.

$\left( \frac{\lambda}{2\pi x^{3}} \right)^{1/2}exp\left( \frac{-\lambda(x-\mu)^{2}}{2\mu^{2}x} \right)$

My book (Fahrmeir & Tutz, Springer) says that

Canonical parameter ($\theta$): $\frac{1}{\mu^{2}}$

Dispersion parameter ($\phi$): $\lambda^{2}$

However my results are a little different from those presented by the author.

$exp\left( \frac{-x/\mu^{2}+2/\mu}{2/\lambda} -\frac{\lambda}{2x}+\frac{ln(\lambda)}{2} - \frac{ln(2\pi x^{3})}{2}\right)$

Canonical parameter ($\theta$): $-\frac{1}{\mu^{2}}$

Dispersion parameter ($\phi$): $2/\lambda$

• I don't think the $\lambda/2x$ term is ancillary. Is it? Mar 8, 2018 at 22:33
• Hi @AdamO, What do you mean by ancillary? As for my new findings, I found out that Fahrmeir uses another parametrization, one that he does not provide, would you know another common way to parametrize the IG distribution? It would help me. Mar 8, 2018 at 23:32

Thus canonical parameter : $-\frac{1}{\mu^2}$
Dispersion: $\frac{2}{\lambda}$
• At the last step, is it not canonically $-1/(2\mu^2)$ and $1/\lambda$ instead? Dec 18, 2022 at 22:01