This paper deals with the problem of estimating an unknown function $f$ in a regression framework with random real-valued design. Our estimation procedure is based on model selection and we consider projection estimators. The models are finite dimensional and nested linear spaces generated by orthonormal bases, particularly trigonometric bases. But instead of expanding directly the target function $f$ on these bases, we rather consider the expansion of $h=f\circ G^{-1}$, where $G$ is the cumulative distribution function of the design. This strategy allows us to compute a projection estimator $\hat{h}$ on a data-selected space, which is easier to compute than least-squares estimators. We establish nonasymptotic mean-squared integrated risk bounds for $\hat{f}=\hat{h}\circ G$ or $\hat{f}=\hat{h}\circ\hat{G}$, depending on whether we assume $G$ is known or not (in this last case, $\hat{G}$ is the empirical distribution function). We study specially adaptive properties for these estimators, in case the regression function belongs to a Besov or Sobolev space.