The Annals of Statistics

On the Stahel-Donoho estimator and depth-weighted means of multivariate data

Yijun Zuo, Hengjian Cui, and Xuming He

Full-text: Open access

Abstract

The depth of multivariate data can be used to construct weighted means as robust estimators of location. The use of projection depth leads to the Stahel-Donoho estimator as a special case. In contrast to maximal depth estimators, the depth-weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature. We also confirm through a finite-sample study that the Stahel-Donoho estimator achieves a desirable balance between robustness and efficiency at Gaussian models.

Article information

Source
Ann. Statist., Volume 32, Number 1 (2004), 167-188.

Dates
First available in Project Euclid: 12 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1079120132

Digital Object Identifier
doi:10.1214/aos/1079120132

Mathematical Reviews number (MathSciNet)
MR2051003

Zentralblatt MATH identifier
1105.62349

Subjects
Primary: 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators
Secondary: 62G35: Robustness 62F35: Robustness and adaptive procedures

Keywords
Asymptotic normality depth breakdown point efficiency projection depth $L$-estimator robustness

Citation

Zuo, Yijun; Cui, Hengjian; He, Xuming. On the Stahel-Donoho estimator and depth-weighted means of multivariate data. Ann. Statist. 32 (2004), no. 1, 167--188. doi:10.1214/aos/1079120132. https://projecteuclid.org/euclid.aos/1079120132


Export citation

References

  • Arcones, M. A. and Giné, E. (1993). Limit theorems for $U$-processes. Ann. Probab. 21 1494--1542.
  • Bai, Z. D. and He, X. (1999). Asymptotic distributions of the maximal depth estimators for regression and multivariate location. Ann. Statist. 27 1616--1637.
  • Chakraborty, B., Chaudhuri, P. and Oja, H. (1998). Operating transformation retransformation on spatial median and angle test. Statist. Sinica 8 767--784.
  • Cui, H. and Tian, Y. (1994). Estimation of the projection absolute median deviation and its application (in Chinese). J. Systems Sci. Math. Sci. 14 63--72.
  • Davies, P. L. (1987). Asymptotic behavior of $S$-estimates of multivariate location parameters and dispersion matrices. Ann. Statist. 15 1269--1292.
  • Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Qualifying paper, Dept. Statistics, Harvard Univ.
  • Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803--1827.
  • Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157--184. Wadsworth, Belmont, CA.
  • Dümbgen, L. (1992). Limit theorem for the simplicial depth. Statist. Probab. Lett. 14 119--128.
  • Gather, U. and Hilker, T. (1997). A note on Tyler's modification of the MAD for the Stahel--Donoho estimator. Ann. Statist. 25 2024--2026.
  • He, X. (1991). A local breakdown property of robust tests in linear regression. J. Multivariate Anal. 38 294--305.
  • He, X. and Wang, G. (1996). Cross-checking using the minimum volume ellipsoid estimator. Statist. Sinica 6 367--374.
  • He, X. and Wang, G. (1997). Convergence of depth contours for multivariate datasets. Ann. Statist. 25 495--504.
  • Kent, J. T. and Tyler, D. E. (1996). Constrained $M$-estimation multivariate location and scatter. Ann. Statist. 24 1346--1370.
  • Kim, J. and Hwang, J. (2001). Asymptotic properties of location estimators based on projection depth. Preprint.
  • Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405--414.
  • Liu, R. Y. (1992). Data depth and multivariate rank tests. In $L_1$-Statistical Analysis and Related Methods (Y. Dodge, ed.) 279--294. North-Holland, Amsterdam.
  • 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.
  • Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion). Ann. Statist. 27 783--858.
  • Maronna, R. A. and Yohai, V. J. (1995). The behavior of the Stahel--Donoho robust multivariate estimator. J. Amer. Statist. Assoc. 90 330--341.
  • Massé, J. C. (1999). Asymptotics for the Tukey depth. Preprint.
  • Mosteller, F. and Turkey, J. W. (1977). Data Analysis and Regression: A Second Course in Statistics. Addison Wesley, Reading, MA.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
  • Rousseeuw, P. J. and Hubert, M. (1999). Regression depth (with discussion). J. Amer. Statist. Assoc. 94 388--433.
  • Rousseeuw, P. J. and Ruts, I. (1998). Constructing the bivariate Tukey median. Statist. Sinica 8 828--839.
  • Stahel, W. A. (1981). Robust estimation: Infinitesimal optimality and covariance matrix estimators. Ph.D. thesis, ETH, Zurich (in German).
  • Tukey, J. W. (1975). Mathematics and the picturing data. In Proceedings of the International Congress of Mathematicians 2 523--531. Canadian Mathematical Congress, Montreal.
  • Tyler, D. E. (1994). Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22 1024--1044.
  • Zhang, J. (1991). Asymptotic theories for the robust PP estimators of the principal components and dispersion matrix. III. Bootstrap confidence sets, bootstrap tests. Systems Sci. Math. Sci. 4 289--301.
  • Zuo, Y. (2003). Projection-based depth functions and associated medians. Ann. Statist. 31 1460--1490.
  • Zuo, Y. and Serfling, R. (2000a). General notions of statistical depth function. Ann. Statist. 28 461--482.
  • Zuo, Y. and Serfling, R. (2000b). Structural properties and convergence results for contours of sample statistical depth functions. Ann. Statist. 28 483--499.