Generalized Linear Models With Unknown Link Functions

Generalized linear models are widely used by data analysts. However, the choice of the link function, i.e., the scale on which the mean is linear in the explanatory variables is often made arbitrarily. Here we permit the data to estimate the link function by incorporating it as an unknown in the model. Since the link function is usually taken to be strictly increasing, by a strictly increasing transformation of its range to the unit interval we can model it as a strictly increasing cumulative distribution function. The transformation results in a domain which is 0,1 as well. We model the cumulative distribution function as a mixture of Beta cumulative distribution functions, noting that the latter family is dense within the collection of all continuous densities on 0,1. For the fitting of the model we take a Bayesian approach, encouraging vague priors, to focus upon the likelihood. We discuss choices of such priors as well as the integrability of the resultant posteriors. Implementation of the Bayesian approach is carried out using sampling based methods, in particular, a tailored Metropolis-within-Gibbs algorithm. An illustrative example utilizing data involving wave damage to cargo ships is provided.