what does the link function of a normal distribution look like?

Question

The link function provides the relationship between the linear predictor and the mean of the distribution function.

In the context of a generalized linear model (GLM)

$\operatorname {E} (\mathbf {Y} )={\boldsymbol {\mu }}=g^{-1}(\mathbf {X} {\boldsymbol {\beta }})$

log linear regression,

$\operatorname {log} (\mathbf {Y} ) = (\mathbf {X} {\boldsymbol {\beta }})$

the link function looks like

g(·) = log(·)

what does the link function of a normal distribution look like?

Answer 1

For a normal distribution, the link function is the identity link

$ g(\mathbf{\mu}) = \mathbf{\mu} = \mathbf{X}\mathbf{\beta}$.

Reference: https://newonlinecourses.science.psu.edu/stat501/node/378/