on fixed design regression models. It allows the application of the mean value theorem for integrals to obtain, for some $\xi_{i,n}\in [x_{i-1,n},x_{i,n}]$, $$f_X(\xi_{i,n})(x_{i,n}-x_{i-1,n})=\int_{x_{i-1,n}}^{x_{i,n}}f_X(x)\,dx=\frac{1}{n}+o\left({\frac 1n}\right)$$ uniformly in $i=1,\dots,n$. Thus it follows from \[X1”\] that $\bar{\Delta}_n$ in assumption \[B1’\] has the exact rate $n^{-1}$ and therefore assumption \[B1’\] reduces to \[B1\]. Further the following Riemann sum approximations for bounded integrable functions $\varphi$ can be applied to get $$\begin{aligned}
\nonumber
\frac 1n \sum_{i=1}^n \varphi(x_{i,n}) &=& \sum_{i=1}^n \varphi(x_{i,n}) (f_X(\xi_{i,n})(x_{i,n}-x_{i-1,n})+o({\textstyle{\frac 1n}})) \\
& =&\int \varphi(x)f_X(x)\,dx+o(1).  \label{riemann}\end{aligned}$$ In the next section we state conditions under which $\hat\vartheta=\arg\min_{\vartheta\in\Theta}M_n(\vartheta)$ consistently estimates $\vartheta_0$.

Main result
===========

To prove consistency of the estimator for the transformation parameter we need the following additional assumptions. Please note that assumption \[M1\] implies identifiability of the transformation $\Lambda_0$ in the class $\mathcal{L}$.

1.  \[M1\] For every $\delta>0$ there exists some $\epsilon>0$ such that $\inf_{\|\t-\t_0\|>\delta} M(\t)\geq\epsilon$.

2.  \[L0\] $\mathcal{L}=\{\Lambda_\t\mid \t\in\Theta\}$ is a class of strictly increasing continuous functions $\mathbb{R}\to\mathbb{R}$.

3.  \[L1\] Let $S=\{\Lambda_0^{-1}(h_0(x))\mid x\in[0,1]\}$. Then the class $\mathcal{L}_S=\{\Lambda_\t|_S\mid \t\in\Theta\}$ is pointwise bounded and uniformly equicontinuous, i.e. $\sup_{\t\in\Theta}|\Lambda_\t(y)|<\infty$ for all $y\in S$, and for every $\epsilon>0$ there exists some $\delta>0$ such that $\sup_{\t\in\Theta}|\Lambda_\t(y)-\Lambda_\t (z)|<\epsilon$ for all $y,z\in S$ with $|y-z|\leq\delta$.

4.  \[L2\] The class $\mathcal{L}_{\tilde S}^1=\{\Lambda_0\circ\Lambda_\t^{-1}|_{\tilde 