Definition for a single predictor

Given

\[(Y_1, X_1), (Y_2, X_2),\dots,(Y_n, X_n) \sim F_{Y, X}\]

It is assumed that the regression function has this form

\[r(x) = E[Y|X=x] = g^{-1}(\beta_0 + \beta_1x) = g^{-1}(\eta)\]

\[g(E[Y|X=x]) = \beta_0 + \beta_1x\]

where \(Y\) is a random variable from a particular distribution of the exponential family, \(g\) is called the link function and \(\eta\) the linear predictor.

So, the model is completely specified given:

  1. The probability distribution of \(Y\).

  2. The relation between the expected values of \(Y\) and a linear combination of the explanatory variables.

Definition for multiple predictors

\[r(\textbf{x}) = E[Y|\textbf{X}=\textbf{x}] = g^{-1}(\beta_1 + \beta_2x_2 + \dotsc + \beta_k x_k ) = g^{-1}(\eta)\]