Optimal rank-based testing for principal components

Annals of Statistics

Volumn 38, Issue 6, 2010, Pages 3245-3299

  Hallin, Marc ^a   Paindaveine, Davy ^a   Verdebout, Thomas ^b  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

^b UNIV LILLE   (France)

Author keywords

Curved experiments; Elliptical densities; Local asymptotic normality; Multivariate ranks and signs; Principal components; Scatter matrix; Shape matrix; Tests for eigenvalues; Tests for eigenvectors

Indexed keywords

EID: 78650116866     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOS810     Document Type: Article

References (37)

1. ANDERSON, T. W. (1963). Asymptotic theory for principal component analysis. Ann. Math. Statist. 34 122-148. MR0145620
2. ANDERSON, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, New York. MR1990662
3. CROUX, C. and HAESBROECK, G. (2000). Principal component analysis based on robust estimators of the covariance or correlation matrix: Influence functions and efficiencies. Biometrika 87 603-618. MR1789812
4. DAVIS, A. W. (1977). Asymptotic theory for principal component analysis: The nonnormal case. Austral. J. Statist. 19 206-212. MR0501590
5. HALLIN, M., OJA, H. and PAINDAVEINE, D. (2006). Semiparametrically efficient rank-based inference for shape II: Optimal R-estimation of shape. Ann. Statist. 34 2757-2789. MR2329466
6. HALLIN, M. and PAINDAVEINE, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103-1133. MR1926170
7. HALLIN, M. and PAINDAVEINE, D. (2004). Rank-based optimal tests of the adequacy of an elliptic VARMA model. Ann. Statist. 32 2642-2678. MR2153998
8. HALLIN, M. and PAINDAVEINE, D. (2005). Affine invariant aligned rank tests for the multivariate general linear model with ARMA errors. J. Multivariate Anal. 93 122-163. MR2119768
9. HALLIN, M. and PAINDAVEINE, D. (2006a). Semiparametrically efficient rank-based inference for shape I: Optimal rank-based tests for sphericity. Ann. Statist. 34 2707-2756. MR2329465
10. HALLIN, M. and PAINDAVEINE, D. (2006b). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 1001-1023. MR2305111
11. HALLIN, M. and PAINDAVEINE, D. (2008a). Optimal rank-based tests for homogeneity of scatter. Ann. Statist. 36 1261-1298. MR2418657
12. HALLIN, M. and PAINDAVEINE, D. (2008b). A general method for constructing pseudo-Gaussian tests. J. Japan Statist. Soc. 38 27-40. MR2458315
13. HALLIN, M. and PURI, M. L. (1994). Aligned rank tests for linear models with autocorrelated errors. J. Multivariate Anal. 50 175-237. MR1293044
14. HALLIN, M. and WERKER, B. J. M. (2003). Semiparametric efficiency, distribution-freeness, and invariance. Bernoulli 9 137-165. MR1963675
15. HETTMANSPERGER, T. P. and RANDLES, R. H. (2002). A practical affine equivariant multivariate median. Biometrika 89 851-860. MR1946515
16. HOTELLING, H. (1933). Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24 417-441.
17. JOLLIFFE, I. T. (1986). Principal Component Analysis. Springer, New York. MR0841268
18. KHURI, A. I. and GOOD, I. J. (1989). The parametrization of orthogonal matrices: A review mainly for statisticians. South African Statist. J. 23 231-250. MR1054642
19. KOLLO, T. and NEUDECKER, H. (1993). Asymptotics of eigenvalues and unit-length eigenvectors of sample variance and correlation matrices. J. Multivariate Anal. 47 283-300. MR1247379
20. KREISS, J. P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112-133. MR0885727
21. LE CAM, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York. MR0856411
22. LE CAM, L. and YANG, G. L. (2000). Asymptotics in Statistics, 2nd ed. Springer, New York. MR1784901
23. MUIRHEAD, R. J. and WATERNAUX, C. M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika 67 31-43. MR0570502
24. PAINDAVEINE, D. (2006). A Chernoff-Savage result for shape. On the nonadmissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206-2220. MR2301635
25. PAINDAVEINE, D. (2008). A canonical definition of shape. Statist. Probab. Lett. 78 2240-2247. MR2458033
26. PEARSON, K. (1901). On lines and planes of closest fit to system of points in space. Philos. Magazine 2 559-572.
27. RAO, C. R. and MITRA, S. K. (1971). Generalized Inverse of Matrices and Applications.Wiley, New York. MR0338013
28. RIEDER, H. (1994). Robust Asymptotic Statistics. Springer, New York. MR1284041
29. SALIBIÁN-BARRERA, M., VAN AELST, S. and WILLEMS, G. (2006). PCA based on multivariate MM-estimators with fast and robust bootstrap. J. Amer. Statist. Assoc. 101 1198-1211. MR2328307
30. SHAPIRO, A. and BROWNE, M. W. (1987). Analysis of covariance structures under elliptical distributions. J. Amer. Statist. Assoc. 82 1092-1097. MR0922173
31. SCHOTT, J. R. (1991). A test for a specific principal component of a correlation matrix. J. Amer. Statist. Assoc. 86 747-751. MR1147100
32. STRASSER, H. (1985). Mathematical Theory of Statistics. de Gruyter, Berlin. MR0812467
33. TYLER, D. E. (1981). Asymptotic inference for eigenvectors. Ann. Statist. 9 725-736. MR0619278
34. TYLER, D. E. (1983). A class of asymptotic tests for principal component vectors. Ann. Statist. 11 1243-1250. MR0720269
35. TYLER, D. E. (1987). A distribution-free M-estimator of multivariate scatter. Ann. Statist. 15 234-251. MR0885734
36. VAN DER VAART, A. (1998). Asymptotic Statistics. Cambridge Univ. Press, Cambridge. MR1652247
37. YANAGIHARA, H., TONDA, T. and MATSUMOTO, C. (2005). The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption. J. Multivariate Anal. 96 237-264. MR2204977