# 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? – AdamO Mar 8 '18 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. – Sergio Andrade Mar 8 '18 at 23:32

Thus canonical parameter : $-\frac{1}{\mu^2}$
Dispersion: $\frac{2}{\lambda}$