Can One Estimate the Unconditional Distribution of Post-Model-Selection Estimators?

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 such as the Akaike information criterion 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 ½ 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.The research of the first author was supported by the Max Kade Foundation and by the Austrian National Science Foundation (FWF), Grant no. P13868-MAT. A preliminary draft of the material in this paper was written in 1999.