# Is the canonical parameter (and therefore the canonical link function) for a Gamma not unique?

Consider $$Y_1, \dots, Y_n$$ independent from the Gamma distribution. For $$y > 0$$: \begin{align} f(y \mid \alpha, \beta) &= \dfrac{1}{\beta^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\beta} \\ &= \exp\left[-\log(\beta^{\alpha}\Gamma(\alpha))+(\alpha-1)\log(y)-\dfrac{y}{\beta} \right] \\ &= \exp\left[y\left(\dfrac{-1}{\beta} \right)-\log(\beta^{\alpha}\Gamma(\alpha))+(\alpha-1)\log(y) \right] \end{align} I define an exponential dispersion family as any distribution whose PMF/PDF is $$f(y \mid \boldsymbol\theta) = \exp\left\{\phi[y\theta - b(\theta)] + c(y, \phi) \right\}\text{, } y \in \Omega$$ where $$\Omega$$ is in the support of a random variable $$Y$$ in the family.

# Method 1:

Therefore, $$Y_i$$ is of the exponential dispersion family with (assuming $$\alpha_i$$ is known) \begin{align} \phi &= 1 \\ \theta_i &= \dfrac{-1}{\beta_i} \\ b(\theta_i) &= \log[\beta_i^{\alpha_i}\Gamma(\alpha_i)] \\ &= \log[(-\theta_i)^{-\alpha_i}\Gamma(\alpha_i)] \\ &= -\alpha_i\log(-\theta_i) + \log \Gamma(\alpha_i) \\ c(\phi, y_i) &= (\alpha_i-1)\log(y_i)\text{.} \end{align} It follows that $$\begin{equation} \mu_i = b^{\prime}(\theta_i) = \dfrac{-\alpha_i}{-\theta_i}(-1) = \dfrac{-\alpha_i}{\theta_i}\text{.} \end{equation}$$ and our canonical link is

$$g(\mu_i) = \theta_i = -\dfrac{\alpha_i}{\mu_i}\tag{*}$$

# Method 2:

Let $$\theta = -\dfrac{1}{\alpha\beta}$$. Then rewrite \begin{align}f(y \mid \alpha, \beta) &= \exp\left[y\left(\dfrac{-1}{\beta} \right)-\log(\beta^{\alpha}\Gamma(\alpha))+(\alpha-1)\log(y) \right] \\ &= \exp\left[y\left(\dfrac{-1}{\beta} \right)-\alpha\log(\beta)-\log\Gamma(\alpha)+(\alpha-1)\log(y) \right] \\ &= \exp\left[y\alpha\theta-\alpha\log(\theta^{-1})-\log\Gamma(\alpha)+(\alpha-1)\log(y) \right] \\ &= \exp\left\{\alpha[y\theta-\log(\theta^{-1})]-\log\Gamma(\alpha)+(\alpha-1)\log(y) \right\} \end{align} in which case, since $$\phi$$ cannot vary depending on $$i$$, \begin{align} \phi &= \alpha \\ \theta_i &= -\dfrac{1}{\alpha\beta_i} \\ b(\theta_i) &= \log(\theta_i^{-1}) = -\log(\theta_i)\\ c(\phi, y_i) &= (\alpha-1)\log(y_i)-\log\Gamma(\alpha)\text{.} \end{align} In this case, $$b^{\prime}(\theta_i) =-\dfrac{1}{\theta_i} = \mu_i$$, which means that $$g(\mu_i) = \theta_i = -\dfrac{1}{\mu_i}\text{.}\tag{**}$$

Wikipedia cites (**) as the canonical link function. Why is (**) preferable over (*) for the canonical link function?

What makes this even more confusing is that http://civil.colorado.edu/~balajir/CVEN6833/lectures/GammaGLM-01.pdf says that the canonical link function is $$\dfrac{1}{\mu_i}$$ (also mentioned in McCullagh and Nelder).

First main question: Are canonical link functions unique, when provided a random component?

Second main question: Is there only supposed to be one canonical link function for each random component? If so, what is wrong with the work above?

The systematic component of the GLM is $$g(\mu_i) = \mathbf{x}_i^{\prime}\boldsymbol\beta\text{.}$$ As long as we do not place any restrictions on the $$\boldsymbol\beta$$ coefficients, without loss of generality, we may ignore all constants (with respect to $$\mu_i$$) of proportionality (these constants will just be absorbed into $$\boldsymbol\beta$$). Thus, any of the link functions above will work as a canonical link function, but $$g(\mu_i) = \dfrac{1}{\mu_i}$$ is the simplest one to work with.