OPTIMAL RANK-BASED TESTING FOR PRINCIPAL COMPONENTS

By Marc Hallin, Davy Paindaveine, Thomas Verdebout. 2010

74 citations recorded by the source corpus.

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,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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b1a68831e3f596f8cade7df89a00dad94c3c248b

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Citations in the corpus

Statistical inference for principal components of spiked covariance matrices (2020, background): “However, the existing literature has only been able to test whether the eigenvectors or eigenspaces of the population covariance matrix are equal to some given ones under the assumption that the dimension is much smaller than the sample size [35, 49, 63, 76, 77].”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, methodology): “…He and Wang (1997), Zuo and He (2006), Zuo and Serfling (2000), among others; see Serfling (2002) for a general introduction on statistical depth, Hallin et al. (2010) for the related concept of quantile, Lòpez-Pintado and Romo (2012) for functional extensions, Zuo (2018) for a state-of-the art…”
ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions (2013, background): “Theoretical analysis in low dimensions is provided by Hallin and Paindaveine (2002a, 2002b, 2004, 2005, 2006), Hallin (Oja); Hallin, Paindaveine, and Verdebout (2010, 2014), and some new extensions to high-dimensional settings are provided in Croux, Filzmoser, and Fritz (2013) and Feng and Liu…”