Regression models form a cornerstone of modern statistical analysis. Generalised linear models (GLM) are popular and widely used models to study the association between an average response and covariates. Recently, probabilistic index models have shown to be a valuable alternative to GLMs when the outcome distribution is skewed or when there are outlying observations. In this project, we unify these two approaches into a single flexible modelling framework, referred to as Unified Regression Models (URM). These URMs also provide a regression framework for the popular and widely applicable class of U-statistics. URMs can further be used for developing novel regression models and this for a variety of settings. The goal of this project is:

1. to work out the estimation theory related to URMs upon using semiparametric theory.
2. to illustrate how URMs can be used to construct new regression models. More specifically, we will work out
   1. a regression model to evaluate the accuracy of a ternary classifier and apply it in the context of radiotherapy.
   2. a powerful omnibus consistent test for the two-sample problem that incorporates auxiliarly baseline information. The test will be applied in the context of randomised clinical trials.

This research will generalise the traditional theory of regression models and U-statisitics, and it will provide a flexible, yet comprehensible, framework for constructing new regression models.