Abstract
Preliminary test estimation is a methodology that combines goodness-of-fit testing and estimation. It is a classical procedure when it is suspected that the parameter to be estimated satisfies some prespecified constraints. 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 $\Sigma_{1},\ldots,\Sigma_{m}$ 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 and an illustration on a real data example is provided.
