Parametric link modification of both tails in binary regression

Common binary regression models such as logistic or probit regression have been extended to include parametric link transformation families. These binary regression models with parametric link are designed to avoid possible link misspecification and improve fit in some data sets. One and two parameter link families have been proposed in the literature (for a review see Stukel (1988)). However in real data examples published so far only one parameter link families have found to improve the fit significantly. This paper introduces a two parameter link family involving the modification of both tails of the link. An analysis based on computationally tractable Bayesian inference involving Monte Carlo sampling algorithms is presented extending earlier work of Czado (1992, 1993b). Finally, the usefulness of the two tailed link modification will be demonstrated in an example where single tail modification can be significantly improved upon by using a two tailed modification.