The Annals of Statistics

Asymptotic distributions of the maximal depth estimators for regression and multivariate location

Zhi-Dong Bai and Xuming He

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We derive the asymptotic distribution ofthe maximal depth regression estimator recently proposed in Rousseeuw and Hubert. The estimator is obtained by maximizing a projection-based depth and the limiting distribution is characterized through a max–min operation of a continuous process. The same techniques can be used to obtain the limiting distribution of some other depth estimators including Tukey’s deepest point based on half-space depth. Results for the special case of two-dimensional problems have been available, but the earlier arguments have relied on some special geometric properties in the low-dimensional space. This paper completes the extension to higher dimensions for both regression and multivariate location models.

Article information

Ann. Statist., Volume 27, Number 5 (1999), 1616-1637.

First available in Project Euclid: 23 September 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G35: Robustness 62F12: Asymptotic properties of estimators
Secondary: 62J05: Linear regression 62H12: Estimation

Asymptotic distribution consistency estimator median multivariate location regression depth robustness


Bai, Zhi-Dong; He, Xuming. Asymptotic distributions of the maximal depth estimators for regression and multivariate location. Ann. Statist. 27 (1999), no. 5, 1616--1637. doi:10.1214/aos/1017939144.

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