Let $Y$ have a Gamma distribution with density

$$f(y ; \alpha, \lambda)=\frac{\lambda^{\alpha} y^{\alpha-1}}{\Gamma(\alpha)} e^{-\lambda y}$$

Show that the Gamma distribution is of exponential dispersion family form. Deduce directly the corresponding expressions for $\mathbb{E}[Y]$ and $\operatorname{Var}[Y]$ in terms of $\alpha$ and $\lambda$. What is the canonical link function?

Let $p<n$. Consider a generalised linear model (g.l.m.) for responses $y_{i}, i=1, \ldots, n$ with random component defined by the Gamma distribution with canonical link $g(\mu)$, so that $g\left(\mu_{i}\right)=\eta_{i}=x_{i}^{\mathrm{T}} \beta$, 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 $i$ th observation, $i=1, \ldots, n$.

Obtain expressions for the score function and Fisher information matrix and explain how these can be used in order to approximate $\hat{\beta}$, the maximum likelihood estimator (m.l.e.) of $\beta$.

[Use the canonical link function and assume that the dispersion parameter is known.]

Finally, obtain an expression for the deviance for a comparison of the full (saturated) model to the g.l.m. with canonical link using the m.l.e. $\hat{\beta}$ (or estimated mean $\hat{\mu}=X \hat{\beta})$.