Generalized linear models: What's the benefit of the underlying theory? I was recently trying to understand generalized linear models (GLMs) and after investing quite a few days, it still hasn't dawned on me what the fundamental benefit of the framework is. Normally, I am used to results like guarantees of convergence, limits for error etc, but all that seems to be missing here.
There is a common framework with underlying distribution, regressors/predictors linear in the coefficients, link functions and finally MLE but it seems to be branching off very quickly into the various subclasses, which each need a separate algebraical and numerical treatment.
So can anyone point me towards what is "general" about the GLMs and what is the benefit of that?
 A: What are the benefits of a unified framework? You are right that we are rapidly going into some much used special cases line logistic regression or Poisson regression, but there is still benefit in having a common framework.

*

*Technology transfer from the general linear model (not generalized!), that is with gaussian errors and identity link. A lot of what one has learned from there can be directly used with glm's, especially all the modeling trick constructing the model matrix $X$.


*A common estimation algorithm (IRLS, iteratively reweighted least squares, see https://stats.stackexchange.com/questions/236676/can-you-give-a-simple-intuitive-explanation-of-irls-method-to-find-the-mle-of-a/237384#237384), leads to a common implementation framework. This is maybe not a mathematical advantage, but software implementation and modeling advantage. It is also a teaching advantage!  This program is not a 100% success, as for some important glm's this algorithm do not work very well, as witnessed by the paper logbin: An R Package for Relative Risk Regression Using the Log-Binomial Model.


*Many common concepts applied to all or most of the special cases, like link function, offsets, variance function, mean function, ... , quasi-likelihood


*One area with little advantage of the common framework is residual analysis, which really need to be studied for each case separately. See for instance family of glm represents the distribution of the response variable or residuals.  But see here for an approach using simulated residuals.


*The Nelder & Wederburn paper introding the glm is here at JSTOR and the original motivations can be found there.
A lot of information can be found at Cross Validated, see as a start https://stats.stackexchange.com/questions/104399/why-do-we-use-glm
