as well as irregular distributions. In what follows, let $\|\cdot\|_{\infty}$ denote the supremum norm and $I\{\cdot\}$ the indicator function.

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

The proof of the lemma is given in section \[proof-consistency-random\] of the appendix. The result applies for a model without transformation. Thus, as a by-product, we show uniform consistency of a boundary curve estimator in models with random covariates (and non-equidistant fixed design, see Lemma \[consistency\]), while in contrast Drees et al. (2018) assumed equidistant design and obtained rates of convergence under stronger assumptions on the error distribution $F_0$ and on the boundary curve $h_0$.

For general $\vartheta\in\Theta$ we define a simple boundary curve estimator accordingly as $$\tilde h_{\vartheta}(x)=\max\{\Lambda_{\vartheta}(Y_{i})|i=1,\ldots,n\text{ with }|X_i-x|\leq b_n\}$$ and it holds that $\tilde h_{\vartheta}=\Lt\circ \Lambda_0^{-1}\circ \tilde h_0$. Thus $\tilde h_\t$ consistently estimates $h_\t$. The local constant estimator can be improved by introducing slight smoothing. To this end, let $K$ be a density with compact support and $a_n$ some sequence of bandwidths that decreases to zero such that $na_n\to \infty$. Define $$\label{hatK}
\hat h_{\vartheta}(x)=\frac{\sum_{i=1}^n \tilde h_{\vartheta}(X_i)K\left(\frac{x-X_i}{a_n}\right)}{\sum_{i=1}^n K\left(\frac{x-X_i}{a_n}\right)},$$ then $\hat h_{\vartheta}$ is also uniformly consistent for $h_\vartheta$; see Lemma \[lemma-smoothing\].

For data as in Example \[example1\], Figures \[graphic3\] 