The Annals of Statistics

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

Juan A. Cuesta-Albertos, Carlos Matrán, and Agustín Mayo-Iscar

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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 n1/2, independently of the rate of convergence of the initial estimator.

Article information

Source
Ann. Statist., Volume 36, Number 5 (2008), 2284-2318.

Dates
First available in Project Euclid: 13 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1223908093

Digital Object Identifier
doi:10.1214/07-AOS541

Mathematical Reviews number (MathSciNet)
MR2458188

Zentralblatt MATH identifier
1148.62038

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62F10: Point estimation 62F12: Asymptotic properties of estimators

Keywords
Multivariate normal distribution elliptical distributions exponential family MVE estimator identifiability censored maximum likelihood truncated maximum likelihood asymptotics breakdown point gross errors model smart estimator

Citation

Cuesta-Albertos, Juan A.; Matrán, Carlos; Mayo-Iscar, Agustín. Trimming and likelihood: Robust location and dispersion estimation in the elliptical model. Ann. Statist. 36 (2008), no. 5, 2284--2318. doi:10.1214/07-AOS541. https://projecteuclid.org/euclid.aos/1223908093


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