the TCM and the TKSCM out-perform in the Model and in the Model , respectively. Nonetheless, in a general setting, we are not able to state which estimator performs better since it depends first on the criterium selected to judge the performance but more importantly on the choice of the bandwidths and the smoothing parameter.

Finally, we recall that the aim of this work is to reduce the dependence between the covariates and the errors. As one can see in Table \[TableCor2paper\], although the estimation of $\vartheta_0$ is less good than expected in the model , the correlations between the covariates and the errors (after transformation) are very small; see also Table \[TableCor1paper\] in the supplement for the correlations in the model . We obtain similar results for the random design case.

Proofs of asymptotic results in the random covariate case
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For the proofs of the asymptotic results let us fix some notation: $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ are the floor and ceiling functions respectively; $\bar{F}=1-F$ denotes the survival function associated to a cdf $F$; $X_1 \overset{d}{=} X_2$ means that two random variables $X_1,X_2$ share the same distribution; $a_n \underset{n \to \infty}{\sim} b_n$ holds if there exists 