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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:

Credit due to http://civil.colorado.edu/~balajir/CVEN6833/lectures/GammaGLM-01.pdf.

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?

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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.

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