the transformed responses $\Lambda_\vartheta (Y_{i})$. Note that those form a boundary regression model with boundary curve $ h_{\vartheta}=\Lt\circ \Lambda_0^{-1}\circ h_0$, because $$\mathbb{P}(\Lambda_\vartheta(Y_{i})\leq h_\vartheta(X_i)\mid X_i=x)=\mathbb{P}(\Lambda_0(Y_{i})\leq h_0(X_i)\mid X_i=x)=1$$ and for each $\delta>0$, $$\mathbb{P}\left(\Lambda_\vartheta(Y_{i})-h_\vartheta(X_i)\leq -\delta\mid X_i=x\right)=\mathbb{P}\left(\Lambda_0(Y_{i})\leq \Lambda_0(\Lambda_\vartheta^{-1}\left(h_\vartheta(x)-\delta)\right)\mid X_i=x\right)<1$$ since $\Delta=h_0(x)-\Lambda_0(\Lambda_\vartheta^{-1}(\Lt(\Lambda_0^{-1}(h_0(x)))  -\delta))>0$ since each $\Lambda_\vartheta$ is strictly increasing. The conditional distribution of $\Lt(Y_{i})$ for some general $\vartheta\in\Theta$ reads as ((Y\_[i]{})yX\_i=x)&=& (\_0(Y\_[i]{})\_0(\^[-1]{}(y))X\_i=x) =F\_0(\_0(\^[-1]{}(y))-h\_0(x)).

\[identifiability\] It is important to give conditions under which the unknown components $\Lambda_0=\Lambda_{\vartheta_0}$, $h_0=h_{\vartheta_0}$ and $F_0$ in model (\[model0-random\]) are identifiable. To this end we impose the following assumptions.

-   Assume that $Y$ has a continuous distribution and w.l.o.g. assume that 0 is in the data range (otherwise shift the data).

-   Assume that $X$ is continuously distributed with support $[0,1]$.

-   Assume $\Lambda_\vartheta(0)=0$ for all $\vartheta\in\Theta$, and $\Lambda_\vartheta$ is strictly increasing and continuous for each $\vartheta\in\Theta$.

-   Assume that if for some $\vartheta_0,\vartheta_1\in\Theta$ one has $$(\Lambda_{\vartheta_1}\circ\Lambda_{\vartheta_0}^{-1})(a-b)=(\Lambda_{\vartheta_1}\circ\Lambda_{\vartheta_0}^{-1})(a)-(\Lambda_{\vartheta_1}\circ\Lambda_{\vartheta_0}^{-1})(b)$$ for all $a,b\in J$, where $J$ is an interval of positive length, then it follows that $\vartheta_0=\vartheta_1$.

-   Assume that $F_0$ (the cdf of $\varepsilon=\Lambda_{\vartheta_0}(Y)-h_{\vartheta_0}(X)$) is strictly increasing.

-   Assume that $h_{\vartheta_0}$ is not constant and is continuous.

Now assume that the model $$\Lambda_\vartheta(Y)=h_\vartheta(X)+\varepsilon(\vartheta)\mbox{ with } X \mbox{ independent from }\varepsilon(\vartheta)$$ holds for $\vartheta=\vartheta_0$ (with 