Abstract :
We consider the problem of estimating the unconditional distribution of a postmodel-
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
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+
