The Generalized Linear Model

[1] What's the Generalized Linear Model?
"The generalized linear model generalizes linear regression by allowing the linear model to be related to the response variable via a link function"
 - Reference : https://en.wikipedia.org/wiki/Generalized_linear_model

In other words, we want to have a model E(Y) as a linear function in the parameters. So beta is a linear form. $g(E(Y))=\beta_{0}+\beta_{1}X_{1}+...+\beta_{p}X_{p}=\mathbb{X}\beta$
Here, g( ) is a link function which generalizes linear regression!

Note that in the General Linear Model, we can predict the beta's by using least squares. However, in the Generalized Linear Model, we use maximum likelihood (*MLE) or Bayesian to predict the beta's.

*Remark) Large -Sample Properties of MLE 
If sample size is large enough and the model is correct, 
MLEs are unbiased, normally distributed and have minimum variance. 
And this MLE is computed by Newton-Raphson algorithm or Fisher scoring.   


[2] What's the Link Function?

Please refer to the reference to find the link function information.
- Reference : http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/logistic-regression/link-function/

[3] Examples of The Generalized Linear Model

Binary Logistic Regression - for more info, click!
 - Logistic Regression Model : $log( \frac{\pi}{1-\pi})= \beta_{0}+ \beta_{1}X_{1}+...+ \beta_{p}X_{p}$
 - Logistic Function : $\pi = \frac{\exp (\beta_{0}+\beta_{1}X_{1}+...+\beta_{p}X_{p})}{1+\exp (\beta_{0}+\beta_{1}X_{1}+...+\beta_{p}X_{p})}$
 where $\pi$ is a probability of success.

- in SAS code: proc logistic

 Poisson Regression

 Gamma Regression