Nonparametric Identification and Estimation of Transformation Models

This paper derives sufficient conditions for nonparametric transformation models to be identified and develops estimators of the identified components. Our nonparametric identification result is global, and is derived under conditions that are substantially weaker than full independence. In particular, we show that a completeness assumption combined with conditional independence with respect to one of the regressors suffices for the model to be identified. The identification result is also constructive in the sense that it yields explicit expressions of the functions of interest. We show how natural estimators can be developed from these expressions, and analyze their theoretical properties. Importantly, it is demonstrated that the proposed estimator of the unknown transformation function converges at the parametric rate.