This 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, asymptotically, 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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Paper provided by Université Libre de Bruxelles, Ecares in its series ECARES Working Papers with number
2009_013.
Find related papers by JEL classification: C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation and Testing
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