The Annals of Statistics

A Class of Asymptotic Tests for Principal Component Vectors

David E. Tyler

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In this paper, the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered. A class of invariant asymptotic tests based on the sample covariance matrix is derived. Tests in this class are shown to be consistent and their local power functions are given. The arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix. The results are shown to generalize when the procedures are based on any affine-invariant $M$-estimate of scatter and when the population is elliptical.

Article information

Ann. Statist., Volume 11, Number 4 (1983), 1243-1250.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62H15: Hypothesis testing
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62H25: Factor analysis and principal components; correspondence analysis 62E20: Asymptotic distribution theory

Elliptical distributions invariance noncentral Wishart robustness spectral decomposition


Tyler, David E. A Class of Asymptotic Tests for Principal Component Vectors. Ann. Statist. 11 (1983), no. 4, 1243--1250. doi:10.1214/aos/1176346337.

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