Statistical Science

Multivariate Nonparametric Tests

Hannu Oja and Ronald H. Randles

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Multivariate nonparametric statistical tests of hypotheses are described for the one-sample location problem, the several-sample location problem and the problem of testing independence between pairs of vectors. These methods are based on affine-invariant spatial sign and spatial rank vectors. They provide affine-invariant multivariate generalizations of the univariate sign test, signed-rank test, Wilcoxon rank sum test, Kruskal–Wallis test, and the Kendall and Spearman correlation tests. While the emphasis is on tests of hypotheses, certain references to associated affine-equivariant estimators are included. Pitman asymptotic efficiencies demonstrate the excellent performance of these methods, particularly in heavy-tailed population settings. Moreover, these methods are easy to compute for data in common dimensions.

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Statist. Sci. Volume 19, Number 4 (2004), 598-605.

First available in Project Euclid: 18 April 2005

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Affine invariance spatial rank spatial sign Pitman efficiency robustness


Oja, Hannu; Randles, Ronald H. Multivariate Nonparametric Tests. Statist. Sci. 19 (2004), no. 4, 598--605. doi:10.1214/088342304000000558.

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  • Blomqvist, N. (1950). On a measure of dependence between two random variables. Ann. Math. Statist. 21 593--600.
  • Chakraborty, B., Chaudhuri, P. and Oja, H. (1998). Operating transformation retransformation on spatial median and angle test. Statist. Sinica 8 767--784.
  • Dümbgen, L. (1998). On Tyler's $M$-functional of scatter in high dimension. Ann. Inst. Statist. Math. 50 471--491.
  • Hallin, M. and Paindaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103--1133.
  • Hand, D. J., Daly, F., Lunn, A. D., McConway, K. J. and Ostrowski, E., eds. (1994). A Handbook of Small Data Sets. Chapman and Hall, London.
  • Hettmansperger, T. P. and Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika 89 851--860.
  • Kendall, M. G. (1938). A new measure of rank correlation. Biometrika 30 81--93.
  • 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.
  • Merchant, J. A., Halprin, G. M., Hudson, A. R., Kilburn, K. H., McKenzie, W. N., Hurst, D. J. and Bermazohn, P. (1975). Responses to cotton dust. Archives of Environmental Health 30 222--229.
  • Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. J. Nonparametr. Statist. 5 201--213.
  • Möttönen, J., Oja, H. and Tienari, J. (1997). On the efficiency of multivariate spatial sign and rank tests. Ann. Statist. 25 542--552.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • Oja, H. (1999). Affine invariant multivariate sign and rank tests and corresponding estimates: A review. Scand. J. Statist. 26 319--343.
  • Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. Wiley, New York.
  • Randles, R. H. (1989). A distribution-free multivariate sign test based on interdirections. J. Amer. Statist. Assoc. 84 1045--1050.
  • Randles, R. H. (2000). A simpler, affine-invariant multivariate, distribution-free sign test. J. Amer. Statist. Assoc. 95 1263--1268.
  • Spearman, C. (1904). The proof and measurement of association between two things. American J. Psychology 15 72--101.
  • Taskinen, S., Kankainen, A. and Oja, H. (2003). Sign test of independence between two random vectors. Statist. Probab. Lett. 62 9--21.
  • Taskinen, S., Oja, H. and Randles, R. H. (2005). Multivariate nonparametric tests of independence. J. Amer. Statist. Assoc. 100. To appear.
  • Tyler, D. E. (1987). A distribution-free $M$-estimator of multivariate scatter. Ann. Statist. 15 234--251.
  • Wilks, S. S. (1935). On the independence of $k$ sets of normally distributed statistical variables. Econometrica 3 309--326.
  • Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Ann. Statist. 28 461--482.