h_{ \vartheta_0} \Vert_{\infty} =o_P(1)$.

[**(ii).**]{} (The fixed design case.) Assume model (\[model0\]) holds under assumptions \[F1’\], \[X1”\], \[X3bis\], \[B1\], \[N1\] and \[M1\]–\[smoothing\]. Then $\hat h_{\hat \vartheta}$ is a consistent estimator of $\ h_{ \vartheta_0}$, i.e. $\Vert \hat h_{\hat \vartheta} - h_{ \vartheta_0} \Vert_{\infty} =o_P(1)$.

[**Proof.**]{} We only prove [**(i)**]{} since the proof of [**(ii)**]{} is identical. Observe first that $$\begin{aligned}
\Vert \hat h_{\hat \vartheta} - h_{ \vartheta_0} \Vert_{\infty} \leq \Vert \hat h_{\hat \vartheta} - h_{ \hat  \vartheta} \Vert_{\infty} + \Vert h_{ \hat \vartheta} - h_{ \vartheta_0} \Vert_{\infty}.\end{aligned}$$ The first term in the right hand side of the above inequality goes to $0$ in probability from Lemma \[lemma-smoothing\] since the consistency holds uniformly over $\vartheta \in \Theta$. Regarding the second term, applying the mean value theorem, there exists some $\vartheta^*(x)$ on the line between $\hat \vartheta$ and $\vartheta_0$ such that

$$\Vert  h_{\hat \vartheta} - h_{  \vartheta_0} \Vert_{\infty} = \sup_{x \in [0,1]} \left| \frac{\partial h_\vartheta (x)^T}{ \partial \vartheta} |_{\vartheta=\vartheta^*(x)} (\hat \vartheta-\vartheta_0) \right|.$$ From Theorem \[theo\], $\hat \vartheta-\vartheta_0=o_P(1)$ which concludes the proof under the assumption \[X3bis\]. $\Box$

Simulations
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To study the small sample behavior, we generate data as $Y=\Lambda_{\vartheta_0}^{-1}(h_0(x)+\varepsilon)$ using the Yeo-Johnson transformation for different values of $\vartheta_0$. We focus on the 