Optimal rank-based testing for principal component
AbstractThis paper provides parametric and rank-based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudo-Gaussian robustifications by Tyler (1981, 1983) and Davis (1977), with which their Gaussian versions are shown to coincide,symptotically, under Gaussian or finite fourth-order moment assumptions, respectively. Such assumptions however restrict the scope to covariance-based principal component analysis. The rank-based tests we are proposing remain valid without such assumptions. Hence, they address a much broader class of problems, where covariance matrices need not exist and principal components are associated with more general scatter matrices. Asymptotic relative efficiencies moreover show that those rank-based tests are quite powerful; when based on van der Waerden or normal scores, they even uniformly dominate the pseudo-Gaussian versions of Anderson’s procedures. The tests we are proposing thus outperform daily practice both from the point of view of validity as from the point of view of efficiency. The main methodological tool throughout is Le Cam’s theory of locally asymptotically normal experiments, in the nonstandard context, however, of a curved parametrization. The results we derive for curved experiments are of independent interest,and likely to apply in other setups.
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Bibliographic InfoPaper provided by ULB -- Universite Libre de Bruxelles in its series Working Papers ECARES with number 2009_013.
Date of creation: 2009
Date of revision:
Publication status: Published by: ECARES
Principal components; Tests for eigenvectors;
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ULB Institutional Repository
2013/2119, ULB -- Universite Libre de Bruxelles.
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- Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2011. "Optimal Rank-Based Tests for Common Principal Components," Working Papers ECARES ECARES 2011-032, ULB -- Universite Libre de Bruxelles.
- Davy Paindaveine & Thomas Verdebout, 2011. "Rank Tests for Elliptical Graphical Modeling," Working Papers ECARES ECARES 2011-039, ULB -- Universite Libre de Bruxelles.
- Marc Hallin & Ramon van den Akker & Bas Werker, 2012.
"Rank-Based Tests of the Cointegrating Rank in Semiparametric Error Correction Models,"
Working Papers ECARES
ECARES 2012-042, ULB -- Universite Libre de Bruxelles.
- Hallin, M. & Akker, R. van den & Werker, B.J.M., 2012. "Rank-based Tests of the Cointegrating Rank in Semiparametric Error Correction Models," Discussion Paper 2012-089, Tilburg University, Center for Economic Research.
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