It is easy to estimate a Poisson regression model using the Newton–Raphson Iterative Technique as it only involves one parameter (mu). However, I am unable to understand how a negative binomial regression model can be estimated using the Newton–Raphson Iterative Technique, as it involve two parameters (mu and dispersion parameter alpha). Can anyone help me in this regard?
The Newton-Raphson technique can be applied to estimating multiple parameters. Instead of finding the gradient of the log likelihood with respect to a single parameter, you find the partial derivatives of the log likelihood with respect to multiple parameters in a vector. The second derivatives are in the form of the Hessian matrix. With the NB distribution both the mu and alpha parameter estimates are updated in each step of the process. Try googling "Newton Raphson multivariate" and you'll find many sites explaining the NR multivariate methodology.