  We consider the estimation of a regression function with random design and
heteroscedastic noise in a nonparametric setting. More precisely, we address
the problem of characterizing the optimal penalty when the regression function
is estimated by using a penalized least-squares model selection method. In this
context, we show the existence of a minimal penalty, defined to be the maximum
level of penalization under which the model selection procedure totally
misbehaves. The optimal penalty is shown to be twice the minimal one and to
satisfy a non-asymptotic pathwise oracle inequality with leading constant
almost one. Finally, the ideal penalty being unknown in general, we propose a
hold-out penalization procedure and show that the latter is asymptotically
optimal.
