linkfcn maps the mean of the response variable mu to
the linear predictor z. linkinv is its inverse.
For the Gaussian family, if the link parameter is positive, then
the extended link is used, defined by $$z =
\frac{sign(\mu)|\mu|^\nu - 1}{\nu}$$ In the other case, the usual Box-Cox link is used.
For the Poisson and gamma families, if the link parameter is
positive, then the link is defined by $$z = \frac{sign(w)
(e^{\nu |w|}-1)}{\nu}$$ where
\(w = \log(\mu)\). In the other case, the usual
Box-Cox link is used.
For the GEV binomial family, the link function is defined by
$$\mu = 1 - \exp\{-\max(0, 1 + \nu z)^{\frac{1}{\nu}}\}$$ for any real \(\nu\). At
\(\nu = 0\) it reduces to the complementary log-log
link.
The Wallace binomial family is a fast approximation to the robit
family. It is defined as $$\mu =
\Phi(\mbox{sign}(z) c(\nu) \sqrt{\nu \log(1 + z^2/\nu)})$$
where \(c(\nu) = (8\nu+1)/(8\nu+3)\)