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

Understand

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).