On the breakdown properties of some multivariate M-functionals

Scandinavian Journal of Statistics

Volumn 32, Issue 2, 2005, Pages 247-264

  Dümbgen, Lutz ^a   Tyler, David E ^b  

^a UNIVERSITY OF BERN   (Switzerland)

^b RUTGERS UNIVERSITY   (United States)

Author keywords

Breakdown; Coplanar contamination; M estimates; M functionals; Scatter matrix; Symmetrization; T distributions; Tight contamination

Indexed keywords

EID: 19744368600     PISSN: 03036898     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1467-9469.2005.00425.x     Document Type: Review

References (29)

1. Adrover, J. G. (1998). Minimax bias-robust estimation of the dispersion matrix of a multivariate distribution. Ann. Statist. 26, 2301-2320.
2. Blake, A. & Marinos, C. (1990). Shape from texture: estimation, isotropy and moments. Art. Intell. 45, 323-380.
3. Davies, P. L. (1987). Asymptotic behavior of S-estimates of multivariate location parameters and dispersion matrices. Ann. Statist. 15, 1269-1292.
4. Dietel, G. (1993). Global location and dispersion functionals. PhD thesis, University of Essen, Germany.
5. Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. PhD qualifying paper, Department of Statistics, Harvard University, Harvard.
6. Donoho, D. L. & Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich Lehmann (eds P. Bickel, K. Doksum & J. L. Hodges, Jr), 157-184. Wadsworth, Belmont, CA.
7. Dümbgen, L. (1998). On Tyler's M-functional of scatter in high dimension. Ann. Inst. Statist. Math. 50, 471-491.
8. Hallin, M. & Paindaveine, D. (2002a). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30, 1103-1133.
9. Hallin, M. & Paindaveine, D. (2002b). Multivariate signed ranks: Randles' interdirections or Tyler's angles? In Statistical data analysis based on the Ll-norm and related methods (ed. Y. Dodge), 271-282. Birkhäuser, Basel.
10. Hettmansperger, T. P. & Randles, R. H. (2003). A practical affine equivariant robust estimate of multivariate location. Biometrika 89, 851-860.
11. Huber, P. J. (1981). Robust statistics. Wiley, New York.
12. Kent, J. T. & Tyler, D. E. (1988). Maximum likelihood estimation for the wrapped Cauchy distribution. J. Appl. Statist. 15, 247-254.
13. Kent, J. T. & Tyler, D. E. (1991). Redescending M-estimates of multivariate location and scatter. Am. Statist. 19, 2102-2119.
14. Kent, J. T. & Tyler, D. E. (1996). Constrained M-estimates of multivariate location and scatter. Ann. Statist. 24, 1346-1370.
15. Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. Ann. Statist. 4, 51-67.
16. Maronna, R. & Yohai, V. J. (1990). The maximum bias of robust covariances. Comm. Statist. Theory Meth. 19, 3925-3933.
17. Maronna, R. & Yohai, V. J. (1993). Bias-robust estimates of regression based on projections. Ann. Statist. 21, 965-990.
18. Maronna, R., Stahel, W. A. & Yohai, V. J. (1992). Bias-robust estimators of multivariate scatter based on projections. J. Mult. Anal. 42, 141-161.
19. Martin, R. D., Yohai, V. J. & Zamar, R. (1989). Min-max robust regression. Ann. Statist. 17, 1608-1630.
20. Randles, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. J. Am. Statist. Assoc. 95, 1263-1268.
21. Rousseeuw, P. J. (1986). Multivariate estimation with high breakdown point. In Mathematical statistics and applications (eds W. Grossman, G. Pflug, I. Vincze & W. Wertz), 283-297. Reidel, Dordrecht.
22. Rousseeuw, P. J. & Croux, C. (1993). Alternatives to the median absolute deviation. J. Am. Statist. Assoc. 88, 1273-1283.
23. Stahel, W. A. (1981). Breakdown of covariance estimators. Research report 31, Fachgruppe Statistik, ETH Zürich.
24. Tatsuoka, K. & Tyler, D. E. (2000). The uniqueness of S and M-functionals under non-elliptical distributions. Ann. Statist. 28, 1219-1243.
25. Tyler, D. E. (1986). Breakdown properties of the M-estimators of multivariate scatter. Technical report, Rutgers University.
26. Tyler, D. E. (1987a). A distribution-free M-estimator of multivariate scatter. Ann. Statist. 15, 234-251.
27. Tyler, D. E. (1987b). Statistical analysis for the angular central Gaussian distribution. Biometrika 74, 579-590.
28. Tyler, D. E. (1994). Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22, 1024-1044.
29. Tyler, D. E. (2002). High breakdown point multivariate M-estimation. Estadistica 54, 227-236.