and \[graphic4\] in the online supplementary material demonstrate the smoothing of the estimator. We use the Epanechnikov-kernel $K(x)=0.75(1-x^2)I_{[-1,1]}(x)$ and bandwidths $b_n=0.5n^{-1/3}$, $a_n=0.5b_n$ with $n=100$.

Based on this estimator we define the joint empirical distribution function of residuals and covariates as $\hat F_{n,\vartheta}(y,s)=\frac 1n\sum_{i=1}^n I\{\Lt(Y_{i})-\hat h_\t(X_i)\leq y\}I\{X_i\leq s\}$. For $\t=\t_0$, the covariate $X_i$ and the error $\Lt(Y_{i})- h_\t(X_i)$ are stochastically independent and thus, the joint empirical distribution function minus the product of the marginals, namely $\hat F_{n,\vartheta}(y,s)-\hat F_{n,\vartheta}(y,1)\hat F_{X,n}(s)$, estimates zero for $\t=\t_0$. Here $\hat F_{X,n}(\cdot)=\hat F_{n,\vartheta}(\infty,\cdot)$ denotes the empirical distribution function of $X_1,\dots,X_n$. We will use this idea to estimate the transformation parameter $\t_0$. To this end, for any function $h:[0,1]\to\mathbb{R}$ define $$\begin{aligned}
\label{G_n-random}
 G_n(\t, h)(y,s)&=& \frac 1n\sum_{i=1}^n I\{\Lt(Y_{i})-h(X_i)\leq y\} \big( I\{X_i\leq s\}-\hat F_{X,n}(s)\big)\end{aligned}$$ and note that $G_n(\t,\hat h_\t)(y,s)=\hat F_{n,\t}(y,s)-\hat F_{n,\t}(y,1)\hat F_{X,n}(s)$. Our criterion function will be $$M_n(\t)= \|G_n(\t,\hat h_\t)\|$$ for some semi-norm $\|\cdot\|$ as described in the following assumption.

1.  \[N1\] $\|\cdot\|$ is a semi-norm such that $\displaystyle\|\Gamma\|\leq c \sup_{y\in C\atop s\in[0,1]}|\Gamma(y,s)|$ for some constant $c>0$ and some compact set $C=[c_1,c_2]\subset\mathbb{R}$ with $c_1,c_2>0$ and $0\in C$, for all measurable functions $\Gamma:\mathbb{R}\times [0,1]\to\mathbb{R}$.

For instance one can consider one of the following semi-norms,

-   $\displaystyle\|\Gamma(y,s)\|=\sup_{s\in [0,1]\atop y\in C} |\Gamma(y,s)|$

-   