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Abstract One-parameter link functions play a fundamental role in regression via generalized linear modelling. This paper develops the general theory for two -parameter links in the very large class of vector generalized linear models by using total derivatives applied to a composite log-likelihood within the Fisher scoring/iteratively reweighted least squares algorithm. We solve a four-decade old problem with an interesting history as our first example: the canonical link for negative binomial regression. The remaining examples are fitting Weibull regression using both the mean and quantile directly compared to GAMLSS, and performing quantile regression based on the Gaussian distribution. Numerical examples based on real and simulated data are given. The methods described here are implemented by the and packages, available on CRAN. Supplementary materials for this article are available online.
Miranda-Soberanis et al. (Thu,) studied this question.