What is the canonical link function for a Tweedie GLM? I was just introduced to the Tweedie distribution (see this or this) but I'm having a hard time finding what the link function is for a Tweedie generalized linear model.
Thoughts?
 A: At your first link it gives:
$\begin{eqnarray*}  \theta &  = &  \left\{  \begin{array}{ll} \frac{\mu ^{1-p}}{1-p} &  p \neq 1 \\ \log \mu &  p = 1 \\ \end{array} \right. \\ \end{eqnarray*}$
$\frac{\mu ^{1-p}}{1-p}$ is indeed the canonical link function for the Tweedie with power parameter $p$. Often (and equivalently, since it only changes the scale and the Tweedie has a scale parameter) just taken to be $\mu^{1-p}$ when $p\neq 1$.
Check: 


*

*$p=0$ (Normal) $\rightarrow$ identity (yep)

*$p=1$ (Poisson) $\rightarrow$ $\log$ (yep, using the limiting case)

*$p=2$ (Gamma) $\rightarrow$ $-$inverse (yep, though often people just say "inverse")

*$p=3$ (inverse Gaussian) $\rightarrow$ $-$inverse$^2$ (yep, up to a scaling constant;
$\hspace {5cm}$again, people often just say "inverse squared")
If you need a reference, see Eqn 2.7 of Ohlsson & Johansson (2006)[1]
[1]: OHLSSON, Esbjörn and JOHANSSON, Björn (2006)
   "Exact Credibility and Tweedie Models,"
ASTIN Bulletin, 36:1,  May, pp 121-133
   DOI: 10.2143/AST.36.1.2014146
pdf 
