$F_0(0)=1$, $F_0(-\Delta)<1$ for all $\Delta>0$.

2.  \[X1’\] The design points $0<x_{1,n}<\dots <x_{n,n}<1$ are deterministic.

3.  \[B1’\] Let $(b_n)_{n \geq 0}$ be a sequence of positive bandwidths that satisfies $\lim_{n\to\infty}b_n=0$ and $\lim_{n \to \infty}\bar{\Delta}_n\log(n)/b_n=0$ for $\bar{\Delta}_n:=\max_{1 \leq i \leq n+1} \left(x_{i,n}-x_{i-1,n}\right)$.

\[consistency\] Under model (\[model0\]) with assumptions \[F1’\], \[X1’\], \[H1\] and \[B1’\] we have $\|\tilde h_0-h_0\|_{\infty}=o_P(1)$.

The proof is given in section \[proof-consistency\] of the online supplementary material. For general $\vartheta\in\Theta$ we define a consistent boundary curve estimator as $$\tilde h_{\vartheta}(x)=\max\{\Lambda_{\vartheta}(Y_{i,n})|i=1,\ldots,n\text{ with }|\tin-x|\leq b_n\} = \Lt(\Lambda_0^{-1}(\tilde h_0(x))).$$ In analogy to (\[G\_n-random\]) we define, for any function $h:[0,1]\to\mathbb{R}$, $$\begin{aligned}
\label{G_n}
 G_n(\t, h)(y,s)&=& \frac 1n\sum_{i=1}^n I\{\Lt(Y_{i,n})-h(\tin)\leq y\} \big( I\{\tin\leq s\}-\hat F_{X,n}(s)\big),\end{aligned}$$ where $$\hat F_{X,n}(s) = \frac 1n \sum_{i=1}^n I\{x_{i,n}\leq s\}.$$ The criterion function is again $M_n(\vartheta)=\|G_n(\vartheta,\hat h_\vartheta)\|$ where the smooth estimator $\hat h_\vartheta$ is defined accordingly as in and with this the transformation parameter estimator is similar to (\[deftheta\]). In order to consider the same deterministic $G$ as in (\[G-random\]) an additional assumption is needed.

1.  \[X1”\] The design points $0<x_{1,n}<\dots <x_{n,n}<1$ are deterministic. There exists a cdf $F_X$ with continuous density function $f_X:[0,1]\to \mathbb{R}$ which is bounded away from zero such that $$\max_{i=1,\dots,n+1}\left|\int_{x_{i-1,n}}^{x_{i,n}}f_X(x)\,dx-{\frac 1n}\right|=o\left({\frac 1n}\right).$$

Assumption \[X1”\] is common in the literature 