General notions of statistical depth function

Annals of Statistics

Volumn 28, Issue 2, 2000, Pages 461-482

  Zuo, Yijun ^a   Serfling, Robert ^b  

^a ARIZONA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF TEXAS AT DALLAS   (United States)

Author keywords

Halfspace depth; Multivariate symmetry; Simplicial depth; Statistical depth functions

Indexed keywords

EID: 0034347419     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1016218226     Document Type: Article

References (48)

1. ARCONES, M. A. and GINÉ, E. (1993). Limit theorems for U-processes. Ann. Probab. 21 1494-1542.
2. BAGGERLY, K. A. and SCOTT, D. W. (1999). Comment on "Multivariate analysis by data depth: Descriptive statistics, graphics and inference," by R. Y. Liu, J. M. Parelius and K. Singh. Ann. Statist. 27 843-844.
3. BARTOSZYŃSKI, R., PEARL, D. K. and LAWRENCE, J. (1997). A multidimensional goodness-of-fit test based on interpoint distances. J. Amer. Statist. Assoc. 92 577-586.
4. BERAN, R. J. and MILLAR, P. W. (1997). Multivariate symmetry models. In Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics (D. Pollard, E. Torgerson and G. L. Yang, eds.) 13-42. Springer, Berlin.
5. CAPLIN, A. and NALEBUFF, B. (1988). On 64%-majority rule. Econometrica 56 787-814.
6. CAPLIN, A. and NALEBUFF, B. (1991a). Aggregation and social choice: A mean voter theorem. Econometrica 59 1-23.
7. CAPLIN, A. and NALEBUFF, B. (1991b). Aggregation and imperfect competition: On the existence of equilibrium. Econometrica 59 25-59.
8. CARRIZOSA, E. (1996). A characterization of halfspace depth. J. Multivariate Anal. 58 21-26.
9. CHAMBERLIN, E. (1937). The Theory of Monopolistic Competition. Harvard Univ. Press.
10. CHEN, Z. (1995). Bounds for the breakdown point of the simplicial median. J. Multivariate Anal. 55 1-13.
11. DONOHO, D. L. (1982). Breakdown properties of multivariate location estimators. Ph. D. qualifying paper, Dept. Statistics, Harvard Univ.
12. DONOHO, D. L. and GASKO, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803-1827.
13. DÜMBGEN, L. (1990). Limit theorems for the empirical simplicial depth. Statist. Probab. Lett. 14 119-128.
14. EDDY, W. F. (1985). Ordering of multivariate data. In Computer Science and Statistics: The Interface (L. Billard, ed.) 25-30. North-Holland, Amsterdam.
15. FRAIMAN, R. and MELOCHE, J. (1996). Multivariate L-estimation. Preprint.
16. 1-Statistical Procedures and Related Topics (Y. Dodge, ed.) 415-430. IMS, Hayward, CA.
17. HE, X. and WANG, G. (1997). Convergence of depth contours for multivariate datasets. Ann. Statist. 25 495-504.
18. HOTELLING, H. (1929). Stability in competition. Econom. J. 39 41-57.
19. KOSHEVOY, G. and MOSLER, K. (1997). Zonoid trimming for multivariate distributions. Ann. Statist. 25 1998-2017.
20. LIU, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414.
21. 1-Statistics and Related Methods (Y. Dodge, ed.) 279-294. North-Holland, Amsterdam.
22. 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.
23. 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.
24. MAHALANOBIS, P. C. (1936). On the generalized distance in statistics. Proc. Nat. Acad. Sci. India 12 49-55.
25. MASSÉ, J. C. and THEODORESCU, R. (1994). Halfplane trimming for bivariate distributions. J. Multivariate Anal. 48 188-202.
26. MIZERA, I. (1998). On depth and deep points: a calculus. Preprint.
27. MOSTELLER, C. F. and TUKEY, J. W. (1977). Data Analysis and Regression. Addison-Wesley, Reading, MA.
28. NIINIMAA, A., OJA, H. and TABLEMAN, M. (1990). On the finite sample breakdown point of the Oja bivariate median and of the corresponding half-samples version. Statist. Probab. Lett. 10 325-328.
29. NOLAN, D. (1992). Asymptotics for multivariate trimming. Stochastic Process. Appl. 42 157-169.
30. OJA, H. (1983). Descriptive statistics for multivariate distributions. Statist. Probab. Lett. 1 327-333.
31. 1 norm in statistical inference. Sankhyā Ser. A 50 289-313.
32. ROUSSEEUW, P. J. and HUBERT, M. (1999). Regression depth (with discussion). J. Amer. Statist. Assoc. 94 388-433.
33. ROUSSEEUW, P. J. and RUTS, I. (1996). Bivariate location depth. J. Roy. Statist. Soc. Ser. C 45 516-526.
34. ROUSSEEUW, P. J. and STRUYF, A. (1998). Computing location depth and regression depth in higher dimensions. Statist. Comput. 8 193-203.
35. RUTS, I. and ROUSSEEUW, P. J. (1996). Computing depth contours of bivariate point clouds. Comput. Statist. Data Anal. 23 153-168.
36. SERFLING, R. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
37. SINGH, K. (1991). A notion of majority depth. Preprint.
38. SMALL, C. G. (1987). Measures of centrality for multivariate and directional distributions. Canad. J. Statist. 15 31-39.
39. SMALL, C. G. (1990). A survey of multidimensional medians. Internat. Statist. Inst. Rev. 58 263-277.
40. STAHEL, W. A. (1981). Robust estimation: infinitesimal optimality and covariance matrix estimators. Ph. D thesis, ETH, Zurich (in German).
41. TUKEY, J. W. (1975). Mathematics and picturing data. In Proceedings of the International Congress on Mathematics (R. D. James, ed.) 2 523-531 Canadian Math. Congress.
42. TYLER, D. E. (1994). Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22 1024-1044.
43. 1-median and associated data depth. Preprint.
44. YEH, A. B. and SINGH, K. (1997). Balanced confidence regions based on Tukey's depth and the bootstrap. J. Roy. Statist. Soc. Ser. B 59 639-652.
45. ZUO, Y. (1999). Affine equivariant multivariate location estimates with best possible breakdown points. Preprint.
46. ZUO, Y. and SERFLING, R. (2000a). Nonparametric notions of multivariate "scatter measure" and "more scattered" based on statistical depth functions. J. Multivariate Anal. To appear.
47. ZUO, Y. and SEKFLING, R. (2000b). Structural properties and convergence results for contours of sample statistical depth functions. Ann. Statist. 28 483-499.
48. ZUO, Y. and SERFLING, R. (2000c). On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. J. Statist. Plann. Inference 84 55-79.