Trimming and likelihood: Robust location and dispersion estimation in the elliptical model

TL;DR

This paper introduces robust location and dispersion estimators in the elliptical model using trimming and likelihood methods, analyzing their properties and robustness, with potential improvements in convergence rates over initial estimators.

Contribution

It proposes new one- or k-step maximum likelihood estimators based on trimmed data, exploring models with truncated, censored, or gross error likelihoods, and studies their theoretical properties.

Findings

Estimators maintain the robustness of initial estimators.

Proposed estimators achieve a convergence rate of n^{1/2}.

Results include existence, uniqueness, and asymptotic properties.

Abstract

Robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated Mahalanobis distance. Here we analyze some one (or $k$)-step Maximum Likelihood Estimators computed on a subsample obtained with such a procedure. We introduce different models which arise naturally from the ways in which the discarded data can be treated, leading to truncated or censored likelihoods, as well as to a likelihood based on an only outliers gross errors model. Results on existence, uniqueness, robustness and asymptotic properties of the proposed estimators are included. A remarkable fact is that the proposed estimators generally keep the breakdown point of the initial (robust) estimators, but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate $n^{1/2}$, independently of the rate of convergence of the initial estimator.