of sinh-arcsinh transformations, see Jones and Pewsey (2009), do shift the location, but they can be modified to fulfill $\Lambda_\vartheta(0)=0$ for all $\vartheta\in\Theta$, e.g. consider $$\Lambda_{(\vartheta_1,\vartheta_2)}(y)=\sinh (\vartheta_1 \sinh^{-1}(y)-\vartheta_2)-\sinh(-\vartheta_2).$$ Here $\vartheta_1>0$ is the tailweight parameter and $\vartheta_2 \in \mathbb{R}$ the skewness parameter. These transformations define also bijective maps $\Lambda_{(\vartheta_1,\vartheta_2)}:\mathbb{R}\to \mathbb{R}$.

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

In the fixed design case we consider a triangular array of independent observations $Y_{i,n}$, $i=1,\ldots,n$, and deterministic design points $0<x_{1,n}<\dots <x_{n,n}<1$. Once again we assume existence of a transformation $\Lambda_0=\Lambda_{\vartheta_0}$ in the class $\mathcal{L}$ such that setting $h_0=h_{\vartheta_0}$ in the corresponding regression model $$\label{model0}
\Lambda_0(Y_{i,n})=h_0(\tin)+\e_{i,n}$$ the cdf of the errors does not depend on the design points, i.e. $\e_{i,n}\sim F_0$ $\forall i,n$. Note that, as in the random design case, we assume $F_0(0)=1$ and $F_0(-\Delta)<1$ for all $\Delta>0$ leading again to $ h_{\vartheta}=\Lt\circ \Lambda_0^{-1}\circ h_0$.

\[identifiability-fixed\] Identifiability can be shown under the same conditions as in Remark \[identifiability\] as long as $ \bar{\Delta}_n:=\max_{1 \leq i \leq n+1} \left(x_{i,n}-x_{i-1,n}\right)\to 0 $; see section \[ident-proof-fixed\] of the supplement.

\[example1\]Figures \[graphic1\] and \[graphic2\] show realizations of the original data and the transformed data using a Yeo and Johnson transformation; see Example \[yeo-johnson\]. For each figure, in the upper left 