$\displaystyle\|\Gamma(y,s)\|=\left(\int \Gamma(y,s)^2w(y,s)\,d(y,s)\right)^{1/2}$ for some integrable weight function $w:\mathbb{R}\times [0,1]\to\mathbb{R}_0^+$ with support included in $C\times [0,1]$

-   $\displaystyle\|\Gamma(y,s)\|=\sup_{s\in [0,1]}\left(\int \Gamma(y,s)^2w(y)\,dy\right)^{1/2}$ for some integrable weight function $w:\mathbb{R}\to\mathbb{R}_0^+$ with support included in $C$

-   $\displaystyle\|\Gamma(y,s)\|=\sup_{y\in C}\left(\int \Gamma(y,s)^2w(s)\,ds\right)^{1/2}$ for some integrable weight function $w:[0,1]\to\mathbb{R}_0^+$.

The first two semi-norms correspond to Kolmogorov-Smirnov and Cramér-von Mises distances, respectively, while the last two are mixtures of both.

Now we define the estimator $\hat \t$ of $\t_0$ as the minimizer of $M_n(\t)$ over $\Theta$, i.e.$$\label{deftheta}
\hat\t=\arg\min_{\t\in\Theta}M_n(\t).$$ For the following theory also the weaker condition $M_n(\hat\vartheta)\leq \inf_{\theta\in\Theta}M_n(\vartheta)+o_P(1)$ is sufficient. Note that ((Y\_[i]{})-h(X\_i)yX\_i=x)&=&(\_0(Y\_[i]{})\_0(\^[-1]{}(y+h(x)))X\_i=x)\
&=&F\_0(\_0(\^[-1]{}(y+h(x)))-h\_0(x)), which reduces to $F_0(y)$ for $\vartheta=\t_0$ and $h=h_0$. Now considering expectations we define $$\begin{aligned}
\label{G-random}
G(\t,h)(y,s)&=&\int F_0\left(\Lambda_0(\Lt^{-1}(y+h(x)))-h_0(x)\right) I\{x\leq s\}f_X(x)\,dx\\
&&{}-\int F_0\left(\Lambda_0(\Lt^{-1}(y+h(x)))-h_0(x)\right)f_X(x)\, dx \, F_X(s)
\nonumber\end{aligned}$$ as deterministic counterpart of $G_n(\t,h)$. Further set $$M(\t)=\|G(\t, h_\t)\|$$ and note that $M(\t_0)=\|G(\t_0,h_{0})\|= 0$. In section 4 we formulate assumptions under which $\hat\vartheta$ consistently estimates $\vartheta_0$.

The fixed design case
---------------------

In the fixed design model (\[model0\]) we define the estimator for the boundary curve $h_0$ as $$\tilde h_0(x)=\max\{\Lambda_0(Y_{i,n})|i=1,\ldots,n\text{ with }|\tin-x|\leq b_n\}$$ and obtain uniform consistency under the following modified assumptions. We set $x_{0,n}=0$ and $x_{n+1,n}=1$.

1.  \[F1’\] Model (\[model0\]) holds with independent $\eps_{1,n},\dots,\eps_{n,n}$ with cdf $F_0$ ($\forall n$) such that 