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We consider the problem of choosing between several models in least-squaresregression with heteroscedastic data. We prove that any penalization procedureis suboptimal when the penalty is a function of the dimension of the model, atleast for some typical heteroscedastic model selection problems. In particular,Mallows' Cp is suboptimal in this framework. On the contrary, optimal modelselection is possible with data-driven penalties such as resampling or $V$-foldpenalties. Therefore, it is worth estimating the shape of the penalty fromdata, even at the price of a higher computational cost. Simulation experimentsillustrate the existence of a trade-off between statistical accuracy andcomputational complexity. As a conclusion, we sketch some rules for choosing apenalty in least-squares regression, depending on what is known about possiblevariations of the noise-level.

Choosing a penalty for model selection in heteroscedastic regression