estimator $\hat h_\vartheta$ to be consistent. While we noticed in the simulations that slight smoothing improves the procedure, the following theorem still holds when $\hat h_\vartheta$ is replaced by the non-smooth estimator $\tilde h_\vartheta$. Assumption \[F2\] holds, e.g. for Hölder-continuous distribution functions $F_0$.

The following theorem states consistency of the transformation parameter estimator.

\[theo\] [**(i).**]{} (The random design case.) Assume model (\[model0-random\]) under assumptions \[F1\]–\[B1\], \[N1\], \[M1\]–\[smoothing\]. Then $\hat\t$ is a consistent estimator, i.e. $\hat\t-\t_0=o_P(1)$.

[**(ii).**]{} (The fixed design case.) Assume model (\[model0\]) under assumptions \[F1’\], \[X1”\], \[H1\], \[B1\], \[N1\], \[M1\]–\[smoothing\]. Then $\hat\t$ is a consistent estimator, i.e. $\hat\t-\t_0=o_P(1)$.

The proof for the random design case is given in section \[proof-theo-random\] of the appendix and the proof for the fixed design case in section \[proof-theo-fixed\] of the supplement. One basic ingredient is the following result, which is proven in section \[proof-lemma-smoothing\] of the appendix for the random design case. The proof for the fixed design case is analogous.

\[lemma-smoothing\] [**(i).**]{} (The random design case.) Under model (\[model0-random\]) with assumptions \[F1\]–\[B1\], \[L0\], \[L1\], \[smoothing\], we have $\sup_{\vartheta\in\Theta}\|\hat h_\vartheta-h_\vartheta\|_\infty=o_P(1)$.

[**(ii).**]{} (The fixed design case.) Under model (\[model0\]) with assumptions \[F1’\], \[X1’\], \[H1\], \[B1’\], \[L0\], \[L1\], \[smoothing\], we have $\sup_{\vartheta\in\Theta}\|\hat h_\vartheta-h_\vartheta\|_\infty=o_P(1)$.

The consistency 