Estimation and goodness-of-fit tests in semi- and nonparametric transformation models

The simple linear regression model is the most commonly used model in statistics when we want to explain the relationship between a dependent variable and a vector of explanatory variables. However, this model relies on heavy assumptions that are not always satisfied in practice, namely that the structure of this model is additive and linear, the variance of the error term is constant and the error term is normally distributed. One way to solve these problems is to transform the response variable. In this thesis, we consider semi- and nonparametric transformation models, i.e., models where the regression function is nonparametric and the transformation of the response is either parametric or nonparametric. The first main objective of this thesis is to construct tests for the parametric form of the regression function in a semiparametric transformation model. We construct a first test based on estimations of the error distribution function and a second test based on estimations of the integrated regression function. A large simulation study is performed to see under which model conditions which test behaves the best. The second main objective of this thesis is to construct new nonparametric estimators for the transformation in a nonparametric transformation model, the first one based on the least squares loss and the second one based on the least absolute deviation loss. The proposed estimators perform better than the estimators already developed in the literature.