On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry

Journal of Statistical Planning and Inference

Volumn 84, Issue 1-2, 2000, Pages 55-79

  Zuo, Yijun ^a   Serfling, Robert ^b  

^a ARIZONA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF TEXAS AT DALLAS   (United States)

Author keywords

Multidimensional median; Multivariate symmetry; Nonparametric location measure; Primary 62H05; Secondary 62G05

Indexed keywords

EID: 0002954623     PISSN: 03783758     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-3758(99)00142-1     Document Type: Article

References (27)

1. Bickel P.J., Lehmann E.L. Descriptive statistics for nonparametric models. II. Location. Ann. Statist. 4:1975;1045-1069.
2. Chakraborty B., Chaudhuri P. On a transformation and re-transformation technique for constructing an affine equivariant multivariate median. Proc. Amer. Math. Soc. 124:1996;2539-2547.
3. Chakraborty B., Chaudhuri P. On an adaptive transformation-retransformation estimate of multivariate location. J. R. Statist. Soc. Ser. B. 60:1998;145-157.
4. Chakraborty, B., Chaudhuri, P., 1999. A note on robustness of multivariate median Statist. Probab. Lett. 41, to appear.
5. Chakraborty B., Chaudhuri P., Oja H. Operating transformation and retransformation on spatial median and angle test. Statist. Sinica. 8:1998;767-784.
6. Chaudhuri P. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91:1996;862-872.
7. Chen Z. Bounds for the breakdown point of the simplicial median. J. Multivariate Anal. 55:1995;1-13.
8. Cover T.M., Efron B. Geometrical probability and random points on a hypersphere. Ann. Math. Statist. 38:1967;213-220.
9. 1 mean. Preprint.
10. Donoho, D.L., 1982. Breakdown properties of multivariate location estimators. Ph.D. Qualifying Paper, Department of Statistics, Harvard University.
11. Donoho D.L., Gasko M. Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20:1992;1803-1827.
12. 1 -norm and Related Methods, North-Holland, Amsterdam, pp. 217-230.
13. Liu R.Y. On a notion of simplicial depth. Proc. Natl. Acad. Sci. USA. 85:1988;1732-1734.
14. Liu R.Y. On a notion of data depth based on random simplices. Ann. Statist. 18:1990;405-414.
15. Lopuhaa H.P., Rousseeuw P.J. Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Statist. 19:1991;229-248.
16. Mosteller, C.F., Tukey, J.W., 1977. Data Analysis and Regression. Addison-Wesley, Reading, MA.
17. Niinimaa A., Oja H., Tableman M. The finite-sample breakdown point of the Oja bivariate median. Statist. Probab. Lett. 10:1990;325-328.
18. Oja H. Descriptive statistics for multivariate distributions. Statist. Probab. Lett. 1:1983;327-333.
19. Randles R.H. A distribution-free multivariate sign test based on interdirections. J. Amer. Statist. Assoc. 84:1989;1045-1050.
20. Randles, R.H., Wolfe, D.A., 1979. Introduction to the Theory of Nonparametric Statistics. Wiley, New York.
21. 1 norm in statistical inference Sankhyā Ser. A. 50:1988;289-313.
22. Small C.G. A survey of multidimensional medians. Internat. Statist. Inst. Rev. 58:1990;263-277.
23. Stahel, W.A., 1981. Robuste Schätzungen: Infinitesimale Optimalität und Schätzungen von Kovarianzmatrizen. Ph.D. Thesis, Eidgenossiche Technische Hochschule, Zürich.
24. Tukey, J.W., 1975. Mathematics and picturing data. Proc. Internat. Congr. Math. Vancouver, vol. 2, 1974, pp. 523-531.
25. Tyler D.E. Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22:1994;1024-1044.
26. Zuo, Y., 1999. Affine equivariant multivariate location estimates with best possible breakdown points, submitted.
27. Zuo, Y., Serfling, R., 1998. General notions of statistical depth function and some related convergence results, submitted.