Exponential Family with Dispersion Parameter Distributions

Question

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

Answer 1

\begin{align*} \left(\frac{\lambda}{2\pi x^3}\right)^{\frac{1}{2}} \exp{\left(-\lambda \frac{(x-\mu)^2}{2\mu^2x}\right)} &= \exp{\left(\ln{\left(\frac{\lambda}{2\pi x^3}\right)^{\frac{1}{2}}}\right)} \exp{\left(-\lambda \frac{(x-\mu)^2}{2\mu^2x}\right)} \\ &= \exp{\left( -\lambda \frac{x^2-2x\mu +\mu^2}{2\,\mu^2x} - \frac{\ln{2\pi x^3}-\ln{\lambda}}{2} \right)}\\ &= \exp{\left( -\frac{\lambda x}{2\mu^2} + \frac{\lambda}{\mu} - \frac{\lambda}{2x} - \frac{\ln{2\pi x^3}-\ln{\lambda}}{2} \right)}\\ &= \exp{\left( \frac{x(-1/\mu^2)+2/\mu}{2/\lambda} - \frac{\lambda}{2x} - \frac{\ln{2\pi x^3}-\ln{\lambda}}{2} \right)} \end{align*}

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

Dispersion: $\frac{2}{\lambda}$