Weighted M-estimators for multivariate clustered data: theory and simulation results

TL;DR

This paper develops weighted M-estimators for multivariate clustered data, providing theoretical properties, robustness analysis, and numerical comparisons to unweighted estimators, highlighting efficiency and robustness trade-offs.

Contribution

It introduces conditions for consistency and asymptotic normality of weighted M-estimators, and analyzes their robustness and efficiency compared to unweighted methods.

Findings

Weighted M-estimators are consistent and asymptotically normal.

Optimal weights increase efficiency but reduce robustness.

Numerical studies compare weighted and unweighted estimators.

Abstract

We study weighted M-estimators for $\mathbb{R}^d$-valued clustered data and give sufficient conditions for their consistency. Their asymptotic normality is established with estimation of the asymptotic covariance matrix. We address the robustness of these estimators in terms of their breakdown point. Comparison with the unweighted case is performed with some numerical studies. They highlight that optimal weights maximizing the relative efficiency have a bad impact on the breakdown point.