  We study the distribution of the adaptive LASSO estimator (Zou (2006)) in
finite samples as well as in the large-sample limit. The large-sample
distributions are derived both for the case where the adaptive LASSO estimator
is tuned to perform conservative model selection as well as for the case where
the tuning results in consistent model selection. We show that the
finite-sample as well as the large-sample distributions are typically highly
non-normal, regardless of the choice of the tuning parameter. The uniform
convergence rate is also obtained, and is shown to be slower than $n^{-1/2}$ in
case the estimator is tuned to perform consistent model selection. In
particular, these results question the statistical relevance of the `oracle'
property of the adaptive LASSO estimator established in Zou (2006). Moreover,
we also provide an impossibility result regarding the estimation of the
distribution function of the adaptive LASSO estimator.The theoretical results,
which are obtained for a regression model with orthogonal design, are
complemented by a Monte Carlo study using non-orthogonal regressors.
