Two Transformation Models and Rank Estimation

In the first half of the paper, we shall investigate the relation of two models both of which have been called a transformation model. One model is in terms of distribution functions and the other is in terms of random variables. We shall show that the former class is larger than the latter and we give an explicit relation between these models. The second half deals with estimation procedures for the regression parameters in the transformation model in terms of distribution functions. After reviewing and extending previously proposed estimators for the model, we derive a new estimator based on ranks. Monte Carlo simulation is performed to compare the empirical properties of several estimators for the Cox model, which is a particular case of our transformation model.