## The Annals of Statistics

### Breakdown properties of location $M$-estimators

#### 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

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

#### 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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