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Affine equivariant multivariate rank methods. (English) Zbl 1011.62053

Summary: The classical multivariate statistical methods (MANOVA, principal components analysis, multivariate multiple regression, canonical correlations, factor analysis, etc.) assume that the data come from a multivariate normal distribution and the derivations are based on the sample covariance matrix. The conventional sample covariance matrix and consequently the standard multivariate techniques based on it are, however, highly sensitive to outlying observations.
In this paper a new, more robust and highly efficient, approach based on an affine equivariant rank covariance matrix is proposed and outlined. The affine equivariant multivariate rank concept is based on the multivariate H. Oja [Stat. Probab. Lett. 1, 327-332 (1983; Zbl 0517.62051)] median.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62J10 Analysis of variance and covariance (ANOVA)
62H20 Measures of association (correlation, canonical correlation, etc.)
62J05 Linear regression; mixed models

Citations:

Zbl 0517.62051
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References:

[1] Anderson, T.W., An introduction to multivariate statistical analysis, (1984), Wiley New York · Zbl 0651.62041
[2] Anderson, T.W., Asymptotic theory for canonical correlation analysis, J. multivariate anal., 70, 1-29, (1999) · Zbl 0943.62056
[3] Baksalary, J.K.; Puntanen, S.; Yanai, H., Canonical correlations associated with symmetric reflexive generalized inverses of the dispersion matrix, Linear algebra appl., 176, 61-74, (1992) · Zbl 0766.62032
[4] Brown, B.M., Hettmansperger, T.P., Möttönen, J., Oja, H., 1997. Rank plots in the affine invariant case In: Dodge, Y. (Ed.), L1-Statistical Procedures and Related Topics, Lecture notes—Monograph series, Institute of Mathematical Statistics, Hayward, CA, pp. 351-362. · Zbl 0933.62004
[5] Croux, C.; Haesbroeck, G., Influence function and efficiency of the minimum covariance determinant scatter matrix estimator, J. multivariate anal., 71, 161-190, (1999) · Zbl 0946.62055
[6] Croux, C.; Haesbroeck, G., Principal component analysis based on robust estimators of the covariance and correlation matrixinfluence functions and efficiencies, Biometrika, 87, 603-618, (2000) · Zbl 0956.62047
[7] Hettmansperger, T.P.; McKean, J.W., Robust nonparametric statistical methods, (1998), Arnold London · Zbl 0887.62056
[8] Hettmansperger, T.P.; Nyblom, J.; Oja, H., On multivariate notions of sign and rank, (), 267-278
[9] Hettmansperger, T.P.; Möttönen, J.; Oja, H., Multivariate affine invariant one-sample signed-rank tests, J. amer. statist. assoc., 92, 1591-1600, (1997) · Zbl 0943.62051
[10] Hettmansperger, T.P.; Möttönen, J.; Oja, H., Affine invariant multivariate rank tests for several samples, Statist. sinica, 8, 765-800, (1998) · Zbl 0905.62062
[11] Hettmansperger, T.P.; Oja, H.; Visuri, S., Discussion of ‘multivariate analysis by data depthdescriptive statistics, graphics and inference’ by Liu et al, Ann. statist., 3, 845-853, (1999)
[12] Isogai, T., Some extension of Haldane’s multivariate Median and its application, Ann. inst. statist. math., 37, 289-301, (1985) · Zbl 0583.62042
[13] Jewell, N.P.; Bloomfield, P., Canonical correlations of past and future for time seriesdefinitions and theory, Ann. statist., 11, 837-847, (1983) · Zbl 0519.62084
[14] Koshevoy, G.A., 2002. Lift-zonoid and multivariate depths. In: Dutter, R., Filzmoser, P., Gather, U., Rousseeuw, P.J. (Eds.), Developments in Robust Statistics, Physica Verlag, Heidelberg, 194-202. · Zbl 05280050
[15] Liu, R.Y; Parelius, J.M.; Singh, K., Multivariate analysis by data depthdescriptive statistics, graphics and inference, Ann. statist., 3, 783-840, (1999)
[16] Malkovich, J.F.; Afifi, A.A., On tests for multivariate normality, J. amer. statist. assoc., 68, 176-179, (1973)
[17] Marden, J.I., Some robust estimates of principal components, Statist. probab. lett., 43, 349-359, (1999) · Zbl 0939.62055
[18] Mardia, K.V., Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519-530, (1970) · Zbl 0214.46302
[19] Mardia, K.V., Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies, Sankhya B, 36, 115-128, (1974) · Zbl 0345.62031
[20] Mardia, K.V.; Kent, J.T.; Bibby, J.M., Multivariate analysis, (1979), Wiley New York · Zbl 0432.62029
[21] Möttönen, J.; Oja, H., Multivariate spatial sign and rank methods, J. nonparam. statist., 5, 201-213, (1995) · Zbl 0857.62056
[22] Möttönen, J.; Hettmansperger, T.P.; Oja, H.; Tienari, J., On the efficiency of affine invariant multivariate rank tests, J. multivariate anal., 66, 108-132, (1998) · Zbl 1127.62361
[23] Oja, H., Descriptive statistics for multivariate distributions, Statist. probab. lett., 1, 327-332, (1983) · Zbl 0517.62051
[24] Oja, H., Affine invariant multivariate sign and rank tests and corresponding estimatesa review, Scand. J. statist., 26, 319-343, (1999) · Zbl 0938.62063
[25] Ollila, E., Croux, C., Oja, H., 2002a. Influence function and asymptotic efficiency of the affine equivariant rank covariance matrix. Conditionally accepted. http://www.cc.jyu.fi/ {}esaolli/ · Zbl 1035.62044
[26] Ollila, E.; Hettmansperger, T.P.; Oja, H., Estimates of the regression coefficients based on the sign covariance matrix, J. Royal Statist. Soc., Ser. B, 447-466, (2002) · Zbl 1090.62052
[27] Ollila, E., Hettmansperger, T.P., Oja, H., 2002c. Affine equivariant multivariate sign methods. Conditionally accepted. http://www.cc.jyu.fi/ {}esaolli · Zbl 1090.62052
[28] Ollila, E., Oja, H., Croux, C., 2002d. The affine equivariant sign covariance matrix: asymptotic behaviour and efficiencies. Conditionally accepted. http://www.cc.jyu.fi/ {}esaolli/ · Zbl 1044.62063
[29] Ollila, E., Koivunen, V., Oja, H., 2003. Estimates of the regression coefficients based on the rank covariance matrix. J. Amer. Statist. Ass., in print. · Zbl 1047.62053
[30] Puntanen, S., On the relative goodness of ordinary least squares estimation in the general linear model. dissertation thesis, Acta univ. tamper. ser. A, 216, (1987)
[31] Puri, M.L.; Sen, P.K., Nonparametric methods in multivariate analysis, (1971), Wiley New York
[32] Rao, C.R., Linear statistical inference and its applications, (1973), Wiley New York · Zbl 0169.21302
[33] Seber, G.A.F., Multivariate observations, (1984), Wiley New York · Zbl 0627.62052
[34] Serfling, R.J., Approximation theorems of mathematical statistics, (1980), Wiley New York · Zbl 0456.60027
[35] Visuri, S., 2001. Array and Multichannel Signal Processing Using Nonparametric Statistics. D.Sc. Thesis, Helsinki University of Technology. http://www.hut.fi/Yksikot/Kirjasto/Diss/2001/isbn951225364X/
[36] Visuri, S.; Koivunen, V.; Oja, H., Sign and rank covariance matrices, J. statist. plann. inference, 91, 557-575, (2000) · Zbl 0965.62049
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