# Preliminary test estimation in ULAN models

## TL;DR

This paper develops a general asymptotic theory for preliminary test estimators within ULAN models, enabling detailed comparisons with classical estimators and illustrating results in both standard and complex modern regression setups.

## Contribution

It establishes a comprehensive asymptotic framework for preliminary test estimators in ULAN models, including modern multisample covariance estimation scenarios.

## Key findings

- Asymptotic behavior of preliminary test estimators is characterized in ULAN models.
- Classical results are recovered in standard linear regression.
- Simulation confirms theoretical predictions.

## Abstract

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.