What is the canonical link function for a Tweedie GLM?

Question

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?

Answer 1

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
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