> [!abstract] > **Exponential dispersion families** are families of distributions indexed by parameters $\theta \in \mathbb{R}, \phi>0$, whose distributions are of the form (among equivalent definitions) $f(y;\theta,\phi)=\exp\left[ \frac{\theta y-\kappa(\theta)}{\phi} \right]\cdot h(y;\phi),$so in particular, with $\phi$ fixed, the family is a 1-parameter [[Exponential Family of Distributions|(natural) exponential family]] with natural parameter $\theta / \phi$. ### Moments of EDFs > [!exposition] Deriving expectation and variance of EDFs > Assuming regularity conditions, differentiating the total mass of EDF densities with respect to $\theta$ gives $\begin{align*} 1&= \int f(y;\theta ,\phi) \, dy\\ 0&= \int \frac{ \partial f }{ \partial \theta } \, dy = \int \frac{y-\kappa'(\theta)}{\phi} \cdot f \, dy \\ &= \frac{1}{\phi} (\mathbb{E}[Y]-\kappa'(\theta))\\ &\Longrightarrow \boxed{\mu(\theta) :=\mathbb{E}_{\theta}[Y]=\kappa'(\theta)}. \end{align*}$Differentiating again gives $\begin{align*} 0&= \int \frac{ \partial^{2} f }{ \partial \theta^{2} } \, dy\\ &= \int \left[ -\frac{\kappa''(\theta)}{\phi} +\left( \frac{y-\kappa'}{\phi} \right)^{2} \right]\cdot f \, dx \\ &= \frac{1}{\phi^{2}}\mathbb{E}_{\theta}\left[ (y-\underbrace{\kappa'(\theta)}_{\mathbb{E}[Y]})^{2}-\phi \kappa''(\theta) \right]\\[0.4em] &\Longrightarrow \boxed{\mathrm{Var}(Y)=\phi \kappa''(\theta)}. \end{align*}$ One consequence is that *$\mu(\theta)$ is monotone increasing hence invertible*: $\mu'(\theta)=\kappa''(\theta)=\frac{\mathrm{Var}(Y)}{\phi}\geq0$since $\phi>0$ for EDFs. This resulting bijection between $\mu$ and $\theta$ gives the [[Link Function#Canonical Link Function|canonical link function]] $g:= (\kappa')^{-1}$ when modeling $Y$ with [[Generalized Linear Models|generalized linear models]]. By equating some model $\eta(\mathbf{x})$ with $g(\mu)=\theta$, *the canonical link allows the GLM to model $\theta$ directly.* Solving for the variance $\mathrm{Var}(Y)$ motivates the **variance function $V$** given by $V(\mu):=\kappa''(\theta(\mu))=\kappa''(\kappa'^{-1}(\mu))=\mathrm{Var}(Y) / \phi.$