result in Theorem \[theo\] should be seen as a first step in the analysis of transformation boundary regression models. An interesting and challenging topic for future research is to derive an asymptotic distribution of $\hat\vartheta-\vartheta_0$ (properly scaled) and to investigate the asymptotic influence of the estimation on subsequent procedures based on the transformed data. This is beyond the scope of the paper as yet there are no results on the uniform asymptotic distribution of $\tilde h_0-h_0$ in the literature.

We finally highlight that under the further condition \[X3bis\] defined below regarding the regularity of the boundary curve, we obtain as a corollary of Theorem \[theo\] the consistency of the estimator $\hat h_{\hat \vartheta}$ of the boundary curve.

1.  \[X3bis\] $\vartheta_0$ is an inner point of a convex parameter space $\Theta$ and $h_\vartheta$ is continuously differentiable with respect to $\vartheta$. Besides, we assume that there exists some $\delta>0$ such that $$\sup_{x \in [0,1]} \sup_{\Vert \vartheta - \vartheta_0 \Vert<\delta}  \left\| \frac{\partial h_\vartheta (x)}{\partial \vartheta} \right\|< \infty.$$

[**(i).**]{} (The random design case.) Assume model (\[model0-random\]) holds under assumptions \[F1\], \[X1\], \[X3bis\], \[B1\], \[N1\] and \[M1\]–\[smoothing\]. Then $\hat h_{\hat \vartheta}$ is a consistent estimator of $ h_{ \vartheta_0}$, i.e. $\Vert \hat h_{\hat \vartheta} - 