Appendix A.5: Computational Details for GLMs for a Canonical Link
Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson · Wiley series in probability and statistics · 2010
Recall that the log-likelihood function for GLMs isFor the canonical link we have r\i = g[£O>/)] = g(ß t ) = Χ/β, and the score equations are ¿)Σ^-^ = » (A-5.1)In matrix form these equations areTo solve the score equations, we can use iteratively reweighted least squares (IRLS), just as we did in the cases of logistic and Poisson regression.We start by finding a first-orderTaylor series approximation in the neighborhood of the solution, η*, which is ¿tot * Λ η-μ,ν^Ι,-'Ι,) Now for a canonical link η ί = 6 h and d Vi ( * \ Generalized Linear Models, Second Edition, by Myers, Montgomery, Vining, and Robinson