  Preliminary test estimation, which is a natural procedure when it is
suspected a priori that the parameter to be estimated might take value in a
submodel of the model at hand, is a classical topic in estimation theory. In
the present paper, we establish general results on the asymptotic behavior of
preliminary test estimators. More precisely, we show that, in uniformly locally
asymptotically normal (ULAN) models, a general asymptotic theory can be derived
for preliminary test estimators based on estimators admitting generic
Bahadur-type representations. This allows for a detailed comparison between
classical estimators and preliminary test estimators in ULAN models. Our
results, that, in standard linear regression models, are shown to reduce to
some classical results, are also illustrated in more modern and involved
setups, such as the multisample one where $m$ covariance matrices
${\pmb\Sigma}_1, \ldots, {\pmb\Sigma}_m$ are to be estimated when it is
suspected that these matrices might be equal, might be proportional, or might
share a common "scale". Simulation results confirm our theoretical findings.
