a constant $ c>0$ such that $\lim_{n \to \infty}a_n/b_n=c$ for two sequences $(a_n)_{n \geq 1}$ and $(b_n)_{n \geq 1}$ of nonnegative numbers; $A^c$ is the complement of a set $A$.

In the following we give the proofs of our results in the random design case whereas the proofs for the fixed design case can be found in the online supplementary material.

Proof of Lemma \[consistency-random\] {#proof-consistency-random}
-------------------------------------

At first we need the following intermediary lemma.

\[lem:model0-random\] Assume model (\[model0-random\]) holds with assumptions \[F1\], \[X1\] and \[B1\]. Then, we have $$\begin{aligned}
\label{Lemma2dot2}
        \sup_{x \in [0,1]}  \min_{ \substack{ i \in \{1,\ldots,n\} \\ |X_{i}-x|\leq b_n }} |\varepsilon_{i}|=o_P(1).
    \end{aligned}$$

[**Proof.**]{} For $n \geq 1$ denote $X_{(1)}<X_{(2)} < \cdots < X_{(n)} $ the order statistics of the random design sample $X_1, X_2,\ldots, X_n$. Let $\pi$ be the random permutation of $\{ 1, \ldots, n\}$ such that $X_{(i)}=X_{\pi(i)}, i=1, \ldots,n$. Due to the independence between the errors and the covariates under \[F1\], $\eps_{\pi(1)},\dots,\eps_{\pi(n)}$ are iid with cdf $F_0$. Let $Z_i=-\eps_{\pi(i)}$, $i=1,\dots,n$, then $Z_1,\dots,Z_n$ are iid with cdf $U$ with $U(x)=1-F_0(-x)$ and we need to show that $$\begin{aligned}
\label{eq1_lem_RD}
    \lim_{n \to \infty} \mathbb{P}\left( \sup_{x \in [0,1]}  \min_{ \substack{ i \in \{1,\ldots,n\} \\ |X_{(i)}-x|\leq b_n }} Z_{i}> \epsilon 