Optimal rank-based tests for Common Principal Components

Bernoulli

Volumn 19, Issue 5 B, 2013, Pages 2524-2556

  Hallin, Marc ^a,b,e   Paindaveine, Davy ^a,c   Verdebout, Thomas ^d  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

^b PRINCETON UNIVERSITY   (United States)

^c SORBONNE UNIVERSITÉ   (France)

^d UNIV LILLE   (France)

^e TILBURG UNIVERSITY   (Netherlands)

Author keywords

Common Principal Components; Local asymptotic normality; Rank based methods; Robustness

Indexed keywords

EID: 84891952407     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/12-BEJ461     Document Type: Article

References (36)

1. Airoldi, J.P. and Hoffmann, R.S. (1984). Age variation in voles (Microtus californicus and Microtus ochrogaster) and its significance for systematic studies. Occasional Papers of the Museum of the Natural History, University of Kansas, Lawrence 111 1-45.
2. Anderson, T.W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley-Interscience [John Wiley & Sons]. MR1990662
3. Bentler, P.M. and Dudgeon, P. (1996). Covariance structure analysis: Statistical practice, theory, and directions. Annu. Rev. Psych 47 563-592.
4. Boente, G. and Orellana, L. (2001). A robust approach to common principal components. In Statistics in Genetics and in the Environmental Sciences (Ascona, 1999). Trends Math. (L.T. Fernholz, S. Morgenthaler and W. Stahel, eds.) 117-145. Basel: Birkhäuser. MR1843174
5. Boente, G. and Orellana, L. (2004). Robust plug-in estimators in proportional scatter models. J. Statist. Plann. Inference 122 95-110. MR2057916
6. Boente, G., Pires, A.M. and Rodrigues, I.M. (2002). Influence functions and outlier detection under the common principal components model: A robust approach. Biometrika 89 861-875. MR1946516
7. Boente, G., Pires, A.M. and Rodrigues, I.M. (2009). Robust tests for the common principal components model. J. Statist. Plann. Inference 139 1332-1347. MR2485129
8. Boik, R.J. (2002). Spectral models for covariance matrices. Biometrika 89 159-182. MR1888370
9. Browne, M.W. (1984). The decomposition of multitrait-multimethod matrices. British J. Math. Statist. Psych. 37 1-21. MR0783496
10. Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics. Ann. Math. Statist. 29 972-994. MR0100322
11. Flury, B. and Riedwyl, H. (1988). Multivariate Statistics: A Practical Approach. New York: Chapman & Hall.
12. Flury, B.K. (1987). Two generalizations of the common principal component model. Biometrika 74 59-69. MR0885919
13. Flury, B.N. (1984). Common principal components in k groups. J. Amer. Statist. Assoc. 79 892-898. MR0770284
14. Flury, B.N. (1986). Asymptotic theory for common principal component analysis. Ann. Statist. 14 418-430. MR0840506
15. Flury, B.N. and Gautschi, W. (1986). An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form. SIAM J. Sci. Statist. Comput. 7 169-184. MR0819466
16. Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707-2756. MR2329465
17. 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
18. Hallin, M. and Paindaveine, D. (2006). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 327-350. MR2305111
19. Hallin, M. and Paindaveine, D. (2008). A general method for constructing pseudo-Gaussian tests. J. Japan Statist. Soc. 38 27-39. MR2458315
20. Hallin, M., Paindaveine, D. and Verdebout, T. (2010). Optimal rank-based testing for principal components. Ann. Statist. 38 3245-3299. MR2766852
21. Hallin, M., Paindaveine, D. and Verdebout, T. (2010). Testing for common principal components under heterokurticity. J. Nonparametr. Stat. 22 879-895. MR2738873
22. Hallin, M. and Werker, B.J.M. (2003). Semi-parametric efficiency, distribution-freeness and invariance. Bernoulli 9 137-165. MR1963675
23. Hettmansperger, T.P. and Randles, R.H. (2002). A practical affine equivariant multivariate median. Biometrika 89 851-860. MR1946515
24. Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. J. Educ. Psych. 24 417-441.
25. Kreiss, J.P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112-133. MR0885727
26. Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. New York: Springer. MR0856411
27. Le Cam, L. and Yang, G.L. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd ed. Springer Series in Statistics. New York: Springer. MR1784901
28. 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
29. Paindaveine, D. (2006). A Chernoff-Savage result for shape: On the non-admissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206-2220. MR2301635
30. Paindaveine, D. (2008). A canonical definition of shape. Statist. Probab. Lett. 78 2240-2247. MR2458033
31. Pearson, K. (1901). On lines and planes of closest fit to system of points in space. Philos. Mag. 2 559-572.
32. Rao, C.R. and Mitra, S.K. (1971). Generalized Inverse of Matrices and Its Applications. New York: Wiley. MR0338013
33. Satorra, A. and Bentler, P.M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. In Proceedings of the Business and Economic Statistics Section of the American Statistical Association 308-313. Alexandria, VA: American Statistical Association.
34. Shapiro, A. and Browne, M.W. (1987). Analysis of covariance structures under elliptical distributions. J. Amer. Statist. Assoc. 82 1092-1097. MR0922173
35. Van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge: Cambridge Univ. Press.
36. Wilks, S.S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Statist. 9 60-62.