标题： 异方差回归中与维度成比例的惩罚项在模型选择中的次优性 显示英文标题

标题： Suboptimality of penalties proportional to the dimension for model selection in heteroscedastic regression

摘要： 我们考虑在异方差数据下的最小二乘回归中选择多个模型的问题。 我们证明，当惩罚与模型的维度成比例时，任何惩罚过程在至少一些典型的异方差模型选择问题中都是次优的。 特别是，在这种框架下，Mallows的$C_p$是次优的，以及任何依赖于数据和其真实分布的“线性”惩罚。 相反，通过数据驱动的惩罚，如$V$-fold 或重采样惩罚（Arlot, 2008a,b），在这种框架下可以实现最优模型选择。 因此，从数据中估计惩罚的“形状”是有用的，即使需要付出更高的计算成本。 显示英文摘要

摘要： We consider the problem of choosing between several models in least-squares regression with heteroscedastic data. We prove that any penalization procedure is suboptimal when the penalty is proportional to the dimension of the model, at least for some typical heteroscedastic model selection problems. In particular, Mallows' $C_p$ is suboptimal in this framework, as well as any "linear" penalty depending on both the data and their true distribution. On the contrary, optimal model selection is possible in this framework with data-driven penalties such as $V$-fold or resampling penalties (Arlot, 2008a,b). Therefore, estimating the "shape" of the penalty from the data is useful, even at the price of a higher computational cost.