Nonparametric multivariate rank tests and their unbiasedness

Bernoulli

Volumn 18, Issue 1, 2012, Pages 229-251

  Jurečková, Jana ^a   Kalina, Jan ^b  

^a CHARLES UNIVERSITY   (Czech Republic)

^b INSTITUTE OF MATHEMATICS   (Czech Republic)

Author keywords

Affine invariance; Contiguity; Kolmogorov Smirnov test; Lehmann alternatives; Liu Singh test; Psi test; Savage test; Two sample multivariate model; Unbiasedness; Wilcoxon test

Indexed keywords

EID: 84856595405     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/10-BEJ326     Document Type: Article

References (43)

1. Amrhein, P. (1995). An example of a two-sided Wilcoxon signed rank test which is not unbiased. Ann. Inst. Statist. Math. 47 167-170.
2. Baringhaus, L. and Franz, C. (2004). On a new multivariate two-sample test. J. Multivariate Anal. 88 190-206. (Pubitemid 38012016)
3. Bickel, P. J. (1969). A distribution free version of the Smirnov two sample test in the p-variate case. Ann. Statist. 40 1-23.
4. Brown, B. M. (1982). Cramér-von Mises distributions and permutation tests. Biometrika 69 619-624.
5. Chaudhuri, P. and Sengupta, D. (1993). Sign tests in multidimension: Inference based on the geometry of the data cloud. J. Amer. Statist. Assoc. 88 1363-1370.
6. Choi, K. and Marden, J. I. (1997). An approach to multivariate rank tests in multivariate analysis of variance. J. Amer. Statist. Assoc. 92 1581-1590.
7. Choi, K. and Marden, J. I. (2005). Tests of multivariate linear models using spatial concordances. J. Nonparametr. Statist. 17 167-185. (Pubitemid 40145068)
8. Friedman, J. H. and Rafsky, L. C. (1979). Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7 697-717.
9. Gibbons, J. D. (1964). A proposed two-sample test and its properties. J. Roy. Statist. Soc. Ser. B 26 305-312.
10. Grose, S. D. and King, M. L. (1991). The locally unbiased two-sided Durbin-Watson test. Econom. Lett. 35 401-407.
11. Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. New York: Academic Press.
12. Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests. New York: Academic Press.
13. Hallin, M. and Pandaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103-1133. (Pubitemid 37095340)
14. Hall, P. and Tajvidi, N. (2002). Permutation tests for equality of distributions in high-dimensional settings. Biometrika 89 359-374. (Pubitemid 41311986)
15. Henze, N. (1988). A multivariate two-sample test based on the number of nearest neighbor type coincidences. Ann. Statist. 16 772-783.
16. Henze, N. and Penrose, M. D. (1999). On the multivariate runs test. Ann. Statist. 27 290-298.
17. Hetmansperger, T. P., Möttönen, J. and Oja, H. (1998). Affine invariant multivariate rank tests for several samples. Statist. Sinica 8 785-800.
18. 1 Norm and Related Methods (Y. Dodge, ed.) 183-189. Basel: Birkhäuser.
19. Jurečková, J. and Milhaud, X. (2003). Derivative in the mean of a density and statistical applications. In Mathematical Statistics and Applications. Festschrift for Constance van Eeden (M. Moore, C. Léger and S. Froda, eds.). IMS Lecture Notes 42 217-232. Beachwood, OH: Inst. Math. Statist.
20. Lehmann, E. L. (1953). The power of rank tests. Ann. Math. Statist. 24 23-42.
21. Liu, R. (1988). On a notion of simplicial depth. Proc. Natl. Acad. Sci. USA 85 1732-1734. (Pubitemid 24315755)
22. Lehmann, E. L. (1997). Testing Statistical Hypotheses, 2nd ed. New York: Springer.
23. Liu, R. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414.
24. Liu, R. and Singh, K. (1993). A quality index based on data depth and multivariate rank tests. J. Amer. Statist. Assoc. 88 252-260.
25. Maa, J.-F., Pearl, D. K. and Bartoszynski, R. (1996). Reducing multidimensional two-sample data to one-dimensional interpoint comparisons. Ann. Statist. 24 1069-1074.
26. Neuhaus, G. and Zhu, L.-X. (1999). Permutation tests for multivariate location problems. J. Multivariate Anal. 69 167-192.
27. Obenchain, R. L. (1971). Multivariate procedures invariant under linear transformations. Ann. Math. Statist. 42 1569-1578.
28. Oja, H. (1987). On permutation tests in multiple regression and analysis of covariance problems. Austr. J. Statist. 29 91-100.
29. Oja, H. (2010). Multivariate Nonparametric Methods with R. An Approach Based on Spatial Signs and Ranks. Lecture Notes in Statistics 199. New York: Springer.
30. Oja, H., Möttönen, J. and Tienari, J. (1997). On the efficiency of multivariate spatial sign and rank tests. Ann. Statist. 25 542-552.
31. Oja, H. and Randles, R. H. (2004). Multivariate nonparametric tests. Statist. Science 19 598-605. (Pubitemid 40687811)
32. Oosterhoff, J. and van Zwet, W. R. (1979). A note on contiguity and Hellinger distance. In Contributions to Statistics: Jaroslav Hájek Memorial Volume (J. Jurečková, ed.) 157-166. Dordrecht: Reidel.
33. Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. New York: Wiley.
34. Randles, R. H. and Peters, D. (1990). Multivariate rank tests for the two-sample location problem. Comm. Statist. Theory Methods 19 4225-4238.
35. Rosenbaum, P. R. (2005). An exact distribution free test comparing two multivariate distributions based on adjacency. J. Roy. Statist. Soc. Ser. B 67 515-530. (Pubitemid 41351403)
36. Savage, I. R. (1956). Contributions to the theory of rank order statistics: The two sample case. Ann. Math. Statist. 27 590-615.
37. Schilling, M. F. (1986). Multivariate two-sample tests based on nearest neighbors. J. Amer. Statist. Assoc. 81 799-806.
38. Sugiura, N. (1965). An example of the two-sided Wilcoxon test which is not unbiased. Ann. Inst. Statist. Math. 17 261-263.
39. Sugiura, N., Murakami, H., Lee, S. K. and Maeda, Y. (2006). Biased and unbiased two-sided Wilcoxon tests for equal sample sizes. Ann. Inst. Statist. Math. 58 93-100.
40. Topchii, A., Tyurin, Y. and Oja, H. (2003). Inference based on the affine invariant multivariate Mann-Whitney-Wilcoxon statistic. J. Nonparametr. Statist. 14 403-414.
41. Tukey, J. W. (1975). Mathematics and the picturing of data. In Proc. Intern. Congress of Mathematicians 2 523-531. Montréal: Canadian Mathematics Congress.
42. Wellner, J. A. (1979). Permutation tests for directional data. Ann. Statist. 7 929-943.
43. Zuo, Y. and He, X. (2006). On the limiting distributions of multivariate depth-based rank sum statistics and related tests. Ann. Statist. 34 2879-2896.