  We investigate the optimality for model selection of the so-called slope
heuristics, $V$-fold cross-validation and $V$-fold penalization in a
heteroscedastic with random design regression context. We consider a new class
of linear models that we call strongly localized bases and that generalize
histograms, piecewise polynomials and compactly supported wavelets. We derive
sharp oracle inequalities that prove the asymptotic optimality of the slope
heuristics---when the optimal penalty shape is known---and $V$ -fold
penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for
a fixed value of $V$ since it recovers asymptotically the oracle learned from a
sample size equal to $1-V^{-1}$ of the original amount of data. Our results are
based on genuine concentration inequalities for the true and empirical excess
risks that are of independent interest. We show in our experiments the good
behavior of the slope heuristics for the selection of linear wavelet models.
Furthermore, $V$-fold cross-validation and $V$-fold penalization have
comparable efficiency.
