McCullagh and Nelder (1989) summarized many approaches to relax the distributional assumptions of the classical linear model under the common term Generalized Linear Models (GLM). A generalized linear model (GLM) is a regression model of the form
An essential feature of the GLM is that the expectation
is directly dependent on a function of
the index
.
Additionally, one assumes that
.
The function
which relates
and
is called the link
function. (Note that
McCullagh and Nelder (1989) actually denote
as the link
function.)
It is easy to see that GLM covers a range of widely used models, e.g.