specific structure of the model, e.g.  linearity of the regression function or homoscedasticity. In practice often a parametric class of transformations is considered from which an ‘optimal’ one should be selected data-dependently (with a specific purpose in mind). A classical example is the class of Box-Cox power transformations introduced for linear models by Box and Cox (1964). There is a vast literature on parametric transformation models in the context of mean regression and we refer to the monograph by Carroll and Ruppert (1988). Powell (1991) introduced Box-Cox transformations in the context of linear quantile regression; see also Mu and He (2007) who considered transformations to obtain a linear quantile regression function. Horowitz (2009) reviewed estimation in transformation models with parametric regression in the cases where either the transformation or the error distribution or both are modeled nonparametrically. Linton, Sperlich and Van Keilegom (2008) suggested parametric estimators for transformations, while the error distribution is estimated nonparametrically and the regression function is additive. In this paper, the aim of the transformation is to induce independence between the covariables and the errors. Linton et al. (2008) considered profile likelihood and minimum distance estimation for the transformation parameter. The results for the profile likelihood 