panel the original data $(x_{i,n},Y_{i,n})$, $i=1,\dots,n=100$, are depicted with their boundary curve, while the upper right panel shows the corresponding non-iid errors. The lower left panel shows the transformed data with the curve $h_0$, while the lower right panel shows the iid errors $\eps_{i,n}$, $i=1,\dots,n$.

Estimating the transformation
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The random design case
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If $\vartheta_0$ were known we could estimate the regression function (upper boundary curve) $h_0$ by a local constant approximation, i.e. $$\label{h_0}
\tilde h_0(x)=\max\{\Lambda_0(Y_{i})|i=1,\ldots,n\text{ with }|X_i-x|\leq b_n\},$$ where $b_n\searrow 0$ is a sequence of bandwidths. For this estimator we will show uniform consistency under the following assumptions.

1.  \[F1\] Model (\[model0-random\]) holds with iid $\eps_{1},\dots,\eps_{n}\sim F_0$ and $F_0(0)=1$, $F_0(-\Delta)<1$ for all $\Delta>0$, and $\eps_{1},\dots,\eps_{n}$ are independent of $X_{1},\dots,X_{n}$.

2.  \[X1\] The covariates $X_1,\dots,X_n$ are iid with cdf $F_X$ and density $f_X$ that is continuous and bounded away from zero on its support $[0,1]$.

3.  \[H1\] The regression function $h_0$ is continuous on $[0,1]$.

4.  \[B1\] Let $(b_n)_{n\in\mathbb{N}}$ be a sequence of positive bandwidths that satisfies $\lim_{n\to\infty}b_n=0$ and $\lim_{n\to\infty}(\log n)/(nb_n)=0$.

Note that we do not require any assumption on the error distribution. In particular, in the setup of regularly varying distributed errors, all the results hold for regular 