标题： 能否估计模型选择后估计量的非条件分布？ 显示英文标题

标题： Can One Estimate The Unconditional Distribution of Post-Model-Selection Estimators?

摘要： 我们研究了估计模型选择后估计量的无条件分布的问题。这里所说的模型选择后的估计量是指这样一个组合过程：首先通过某种模型选择准则（例如AIC）或假设检验程序选择一个模型，然后基于同一个数据集利用最小二乘法或最大似然法等方法估计所选模型中的参数。我们证明了即使在渐近情况下，也很难以合理精度估计这个无条件分布。具体来说，我们证明了对于该分布的任何估计量都不可能是一致的（甚至在局部也不一致）。这一结论是基于分布估计量的（局部）极小极大下界推导而来的；此处性能由估计误差超过给定阈值的概率来衡量。这些下界在大样本情况下会趋近于1/2甚至1，具体取决于所考虑的情况。类似的结果也适用于模型选择后估计量的线性函数（例如预测变量）的分布。 显示英文摘要

摘要： We consider the problem of estimating the unconditional distribution of a post-model-selection estimator. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion like AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate the unconditional distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for the distribution; performance is here measured by the probability that the estimation error exceeds a given threshold. These lower bounds are shown to approach 1/2 or even 1 in large samples, depending on the situation considered. Similar impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the post-model-selection estimator.