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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"Semiparametric efficiency, distribution-freeness and invariance,"
Open Access publications from Tilburg University
urn:nbn:nl:ui:12-117062, Tilburg University.
- Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
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2012-089, Tilburg University, Center for Economic Research.
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