The Annals of Statistics

Convergence of depth contours for multivariate datasets

Xuming He and Gang Wang

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Contours of depth often provide a good geometrical understanding of the structure of a multivariate dataset. They are also useful in robust statistics in connection with generalized medians and data ordering. If the data constitute a random sample from a spherical or elliptic distribution, the depth contours are generally required to converge to spherical or elliptical shapes. We consider contour constructions based on a notion of data depth and prove a uniform contour convergence theorem under verifiable conditions on the depth measure. Applications to several existing depth measures discussed in the literature are also considered.

Article information

Ann. Statist., Volume 25, Number 2 (1997), 495-504.

First available in Project Euclid: 12 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62F35: Robustness and adaptive procedures 62H05: Characterization and structure theory 60H05: Stochastic integrals

Convergence contour data depth elliptic distributions location-scatter $M$-estimator multivariate dataset robustness


He, Xuming; Wang, Gang. Convergence of depth contours for multivariate datasets. Ann. Statist. 25 (1997), no. 2, 495--504. doi:10.1214/aos/1031833661.

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  • Arcones, M. A. and Gin´e, E. (1993). Limit theorems for U-processes. Ann. Probab. 21 1494-1542.
  • Arcones, M. A., Chen, Z. and Gin´e, E. (1994). Estimators related to U-processes with applications to multivariate medians: asy mptotic normality. Ann. Statist. 22 1460-1477.
  • Beran, R. (1996). Confidence sets centered at Cp-estimators. Ann. Inst. Statist. Math. 48. To appear.
  • Bose, A. and Chaudhuri, P. (1993). On the dispersion of multivariate median. Ann. Inst. Statist. Math. 45 541-550.
  • Davies, P. L. (1987). Asy mptotic behavior of S-estimates of multivariate location parameters and dispersion matrices. Ann. Statist. 15 1269-1292.
  • Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803-1827.
  • Huber, P. J. (1981). Robust Statistics. Wiley, New York.
  • Koshevoy, G. and Mosler, K. (1996). Zonoid trimming for multivariate distributions. Preprint.
  • Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414.
  • Liu, R. Y. and Singh, K. (1993). A quality index based on data depth and multivariate rank tests. J. Amer. Statist. Assoc. 88 252-260.
  • Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. Ann. Statist. 4 51-67.
  • Mass´e, J. C. and Theodorescu, R. (1994). Halfplane trimming for bivariate distributions. J. Multivariate Anal. 48 188-202.
  • Niinimaa, A., Oja, H. and Tableman, M. (1990). The finite-sample breakdown point of the Oja bivariate median and of the corresponding half-samples version. Statist. Probab. Lett. 10 325-328.
  • Nolan, D. (1992). Asy mptotics for multivariate trimming. Stochastic Process. Appl. 42 157-169.
  • Oja, H. (1983). Descriptive statistics for multivariate distributions. Statist. Probab. Lett. 1 327- 332.
  • Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. Wiley, New York.
  • Tukey, J. W. (1975). Mathematics and picturing data. Proceedings of International Congress of Mathematicians, Vancouver 2 523-531.
  • Vapnik, V. N. and Cervonenkis, A. Ya. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16 264-280.