assumption is met, e.g., by Müller and Wefelmeyer (2010), Meister and Rei[ß]{} (2013), and Drees et al. (2018). A transformation inducing (approximate) independence between the covariable and the error would allow for a global bandwidth selection in the adaptive regression estimator suggested by Jirak et al. (2014). Wilson (2003) pointed out that in production frontier models, independence assumptions are needed for validity of bootstrap procedures for nonparametric frontier models (see Simar and Wilson, 1998) and suggested some tests for independence of errors and covariates (see also Drees et al., 2018).

While Linton et al. (2008) found advantages of the profile likelihood approach over minimum distance estimation of the transformation parameter in corresponding mean regression transformation models, this is at the cost of strong regularity conditions, among others a bounded error density with bounded derivative. In the context of boundary models with error distribution which is regularly varying at zero and irregular, one needs to avoid assumptions on bounded densities. Thus we investigate a minimum distance approach to estimate the transformation parameter and give mild model assumptions under which the estimator is consistent. We consider the cases of random covariates and deterministic (fixed) design points, which are both meaningful. The equidistant 