6.8 The quasi-Poisson model

This model (Hoef and Boveng 2007) is a generalization of Poisson model where we can allow the variance to linearly depends on the mean \(\lambda\) by:

\[\begin{equation} Var(y)=\theta \lambda \tag{6.73} \end{equation}\]

Then, we estimate the regression function @(eq:count-poisson-lambda) using the IWLS iterative weighted least squares, which is the WLS \(\beta_{WLS}=\bigg(X^tW^{-1}X\bigg)^{-1}X^tW^{-1}y\) discussed in @(weighted-least-squares-wls) used iteratively. In our case the matrix \(W\) is a diagonal matrix given by:

\[\begin{equation} W=diag\Bigg(\frac{\lambda_1^2}{\theta\lambda_1}...\frac{\lambda_n^2}{\theta\lambda_n}\Bigg)=diag\Bigg(\frac{\lambda_1}{\theta}...\frac{\lambda_n}{\theta}\Bigg) \tag{6.74} \end{equation}\]