  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.
