On page 12 of the GLM chapter the structure in matrix form is given as Y = inverse link (XBEta + error)+error which makes sense. Then on page 21 on matrix inversion the inverse link function is removed because we are considering an inverse normal with identity link, which makes sense. Then I can follow all the hat matrix stuff from there. It is detailed that when more complex link functions used we have to use Newton raphson. So is calculating the estimates using hat matrix directly only appropriate for identity link? What happened to all the maximum likelihood stuff from ct6? That seemed much more flexible
You're quite right George, it would be lovely to use MLE's to arrive at an answer directly. The problem is, we can't solve these other cases algebraically. Instead we have to use the Newton Raphson method, and repeat this iteratively until we arrive at solution with the required degree of accuracy.
No, it's because the MLE's cannot be obtained in closed form. Why not try it? Have a go at finding the MLE's for the parameters of the gamma distribution. You'll end up doing lots of maths, but you'll never be able to solve the partial differential equations.
