In GLM, assuming a scalar $Y$ and $\theta$ for the underlying distribution with p.d.f.
$$f_Y(y | \theta, \tau) = h(y,\tau) \exp{\left(\frac{\theta y – A(\theta)}{d(\tau)} \right)}$$
It can be shown that $ \mu = \operatorname{E}(Y) = A'(\theta)$. If the link function $g(\cdot)$ satisfies the following, $$g(\mu)=\theta = X'\beta $$ where $X'\beta$ is the linear predictor, then $g(\cdot)$ is called the canonical link function for this model.
My question is, does a canonical link function always exist for a GLM? In other words, can $A'(\theta)$ always be inverted? What are the necessary conditions for a canonical link function to exist?
Best Answer
For these distributions $A'(\theta) = E(Y)$ and $A''(\theta)=Var(Y)/d(\tau)$
Since the variance and dispersion parameter are non-zero (and even positive) $A'(\theta)$ is a strictly increasing function and must be invertible.
However, I am not sure if there are distributions of this family that have an infinite variance. I was not able to find such examples.