\mbox{ for all } \epsilon>0\end{aligned}$$ (see, e.g., Lemma 9.21 in Kosorok, 2008).

-   \[L1\]–\[F2\] can be seen as minimal assumptions on the class $\mathcal{L}=\{\Lambda_\vartheta\mid\vartheta\in\Theta\}$ and $F_0$. As typically the sets $S$, $\tilde S$ and $\tilde C$ are unknown, the assumptions can be replaced by stronger assumptions that hold on all compact sets. Besides, working on compact set transformation parameter, assumptions \[L1\]–\[L2\] hold for most of transformations used in practice such as the Box and Cox transformations (see Box and Cox, 1964) (suitably modified taking into account the data range), the exponential transformations (see Manly, 1976), the sinh-arcsinh transformations (see Jones and Pewsey, 2009). For instance, with regard to Yeo-Johnson transformations, when $\vartheta \in \Theta=[0,2]$, $\Lambda_{\vartheta}: \mathbb{R}\to \mathbb{R}$ defines a bijective map (see Remark \[yeo-johnson\]) and both $\Lambda_\t$ and $\Lambda_\t^{-1}$ have uniform bounded derivatives on compact sets so that one may show that they fulfill assumptions \[L1\]–\[L2\] using the mean value theorem to $\Lambda_\t$ and $\Lambda_\t^{-1}$. Further under stronger assumptions on the smoothness of $F_0$, $\Lambda_\t$ and $\Lambda_0\circ\Lambda_\t^{-1}$, the theoretical results can be generalized to semi-norms that are not restricted to a compact $C\times[0,1]$ as in assumption \[N1\].

-   \[smoothing\] is standard in kernel smoothing and is needed for the smoothed 