The Annals of Statistics

Breakdown properties of location $M$-estimators

Guoying Li and Jian Zhang

Full-text: Open access

Abstract

In this article, we consider the asymptotic behavior of three kinds of sample breakdown points. It is shown that for the location $M$-estimator with bounded objective function, both the addition sample breakdown point and the simplified replacement sample breakdown point strongly converge to the gross-error asymptotic breakdown point, whereas the replacement sample breakdown point strongly converges to a smaller value. In addition, it is proved that under some regularity conditions these sample breakdown points are asymptotically normal. The addition sample breakdown point has a smaller asymptotic variance than the simplified replacement sample breakdown point. For the commonly used redescending $M$-estimators of location, numerical results indicate that among the three kinds of sample breakdown points, the replacement sample breakdown point has the largest asymptotic variance.

Article information

Source
Ann. Statist., Volume 26, Number 3 (1998), 1170-1189.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1024691093

Digital Object Identifier
doi:10.1214/aos/1024691093

Mathematical Reviews number (MathSciNet)
MR1635381

Zentralblatt MATH identifier
0929.62031

Subjects
Primary: 62F35: Robustness and adaptive procedures

Keywords
Sample breakdown point redescending $M$-estimator asymptotics

Citation

Zhang, Jian; Li, Guoying. Breakdown properties of location $M$-estimators. Ann. Statist. 26 (1998), no. 3, 1170--1189. doi:10.1214/aos/1024691093. https://projecteuclid.org/euclid.aos/1024691093


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