Let $f_{0}$ be a probability density function, with cumulant generating function $K$. Define what it means for a random variable $Y$ to have a model function of exponential dispersion family form, generated by $f_{0}$.

A random variable $Y$ is said to have an inverse Gaussian distribution, with parameters $\phi$ and $\lambda$ (both positive), if its density function is

$$f(y ; \phi, \lambda)=\frac{\sqrt{\lambda}}{\sqrt{2 \pi y^{3}}} e^{\sqrt{\lambda \phi}} \exp \left\{-\frac{1}{2}\left(\frac{\lambda}{y}+\phi y\right)\right\} \quad(y>0)$$

Show that the family of all inverse Gaussian distributions for $Y$ is of exponential dispersion family form. Deduce directly the corresponding expressions for $E(Y)$ and $\operatorname{Var}(Y)$ in terms of $\phi$ and $\lambda$. What are the corresponding canonical link function and variance function?

Consider a generalized linear model, $M$, for independent variables $Y_{i}(i=1, \ldots, n)$, whose random component is defined by the inverse Gaussian distribution with link function $g(\mu)=\log (\mu):$ thus $g\left(\mu_{i}\right)=x_{i}^{\mathrm{T}} \beta$, where $\beta=\left(\beta_{1}, \ldots, \beta_{p}\right)^{\mathrm{T}}$ is the vector of unknown regression coefficients and $x_{i}=\left(x_{i 1}, \ldots, x_{i p}\right)^{\mathrm{T}}$ is the vector of known values of the explanatory variables for the $i^{\text {th }}$observation. The vectors $x_{i}(i=1, \ldots, n)$ are linearly independent. Assuming that the dispersion parameter is known, obtain expressions for the score function and Fisher information matrix for $\beta$. Explain how these can be used to compute the maximum likelihood estimate $\widehat{\beta}$ of $\beta$.