The Generalized Linear Regression Model

Abstract

Consider the linear regression model that represents the statistical dependence of study variable y on K explanatory variables X 1 ,. .. , X K and random error y = Xβ + (4.1) with the following assumptions: (i) E() = 0, (ii) E() = σ 2 W where W is positive definite, (iii) X is a nonstochastic matrix and (iv) rank(X) = K. This is termed as generalized linear regression model or generalized linear model. Note that in the classical regression model, E() = σ 2 I. If E() = σ 2 W where W is a known positive definite matrix, the generalized linear model can be reduced to the classical model: Because W is positive definite, so W has a positive definite inverse W −1. According to theorems (cf. Theorem A.41), product representations exist for W and W −1 : W = M M, W −1 = N N where M and N are the square and regular matrices. Thus (N N) = (M M) −1 , including N M M N = N W N = I. If the generalized linear model y = Xβ + is transformed by multiplication from the left with N , the trans