random design case
----------------------

Consider independent and identically distributed observations $(X_i,Y_i)$, $i=1,\ldots,n$, with the same distribution as $(X,Y)$, where $Y$ is univariate and $X$ is distributed on $[0,1]$. Further consider a family $\mathcal{L}=\{\Lt|\vartheta\in\Theta\}$ of strictly increasing and continuous transformations. Throughout the paper we assume existence of a transformation $\Lambda_{\vartheta_0}$ in the class $\mathcal{L}$ such that in the corresponding boundary regression model $$\label{model0-random}
\Lambda_{\vartheta_0}(Y)=h_{\vartheta_0}(X)+\e$$ the errors and the covariates are stochastically independent. Note that for notational simplicity we set $\Lambda_0=\Lambda_{\vartheta_0}$ and $h_0=h_{\vartheta_0}$. Further denote by $F_0$ the cumulative distribution function (cdf) of the independent and identically distributed (iid) $\e_i=\Lambda_0(Y_i)-h_0(X_i)$, $i=1,\dots,n$. Then we assume that $F_0(0)=1$ and $F_0(-\Delta)<1$ for all $\Delta>0$. This identifies the function $h_0$ as the upper boundary curve of the observations since $$\begin{aligned}
\mathbb{P}(\Lambda_0(Y_{i}) \leq h_0(X_i)\mid X_i=x)&=&1  \mbox{ for all } x\in[0,1]\\
\mathbb{P}(\Lambda_0(Y_{i}) -h_0(X_i)\leq -\Delta\mid X_i=x)&<&1 \mbox{ for all }x\in [0,1], \Delta>0. $$ The aim is to estimate $\vartheta_0$ from the observations.

Note that even if the model does not hold exactly (i.e. there does not exist any $\vartheta_0\in \Theta$ that leads to exact independence of the errors and covariates) the transformation can be useful in applications because it will reduce the dependence.

For each $\vartheta\in\Theta$ one can consider 