our notations $\Lambda_{\vartheta_0}=\Lambda_0$, $h_{\vartheta_0}=h_0$, $\varepsilon(\vartheta_0)=\varepsilon$) and for $\vartheta=\vartheta_1$.

Note that from the assumption it follows that $\mathbb{P}(\varepsilon(\vartheta)\leq 0)=1$, $\mathbb{P}(\varepsilon(\vartheta)\leq -\Delta)<1$ for each $\Delta>0$, such that $h_\vartheta$ is the upper boundary curve in the model (for $\vartheta\in \{\vartheta_0,\vartheta_1\}$).

We show in section \[ident-proof\] of the appendix that it follows that $\vartheta_0=\vartheta_1$. Thus the transformation is identifiable. Further $h_{\vartheta_0}(x)$ is then the right endpoint of the conditional distribution of $\Lambda_{\vartheta_0}(Y)$, given $X=x$, and $F_0$ is identified as cdf of $\Lambda_{\vartheta_0}(Y)-h_{\vartheta_0}(X)$.

If the function class $\mathcal{L}$ contains the identity, then the assumptions rule out that it contains transformations which are linear on some interval with positive length. On the other hand it is clear that linear transformations can never reduce the dependence between the covariate and the error distribution.

\[yeo-johnson\] In this example we give two classes of transformations that fulfill the identifiability assumptions.

Yeo and Johnson (2000) generalized the Box-Cox transformations by suggesting $$\Lambda_\vartheta(y) = \begin{cases} \frac{(y+1)^\vartheta-1}{\vartheta}, &\text{if } y \ge 0,\vartheta\neq 0 \\
 \log(y+1), &\text{if } y\ge 0,\vartheta=0\\
 -\frac{(-y+1)^{2-\vartheta}-1}{2-\vartheta}, &\text{if } y<0,\vartheta\neq 2 \\
 -\log(-y+1), &\text{if } y<0,\vartheta=2,
 \end{cases}$$ which are typically considered for $\t\in\Theta=[0,2]$ because then they are bijective maps $\Lambda_\t:\mathbb{R}\to \mathbb{R}$. Note that $\Lambda_\vartheta(0)=0$ for all $\vartheta\in\Theta$.

The class 