Asymptotic theory for common principal component analysis. (English) Zbl 0613.62075

Flury, Bernard N.

Ann. Stat. 14, 418-430 (1986).

Common principal component analysis is a generalization of principal component analysis to k groups, and it is assumed that the \(p\times p\) covariance matrices \(\Sigma_ 1,...,\Sigma_ k\) of k populations can be diagonalized by the same orthogonal transformation, i.e., there exists an orthogonal matrix \(\beta\) such that \(\beta '\Sigma_ i\beta =\Lambda_ i\) is diagonal \((i=1,...,k)\). The asymptotic distribution of the maximum likelihood estimates of \(\beta\) and \(\Lambda_ i\) are derived. In particular, if \(\lambda_{ij}\) \((j=1,...,p)\) are the diagonal elements of \(\Lambda_ i\), and \({\hat \lambda}_{ij}\) are their maximum likelihood estimates, then the statistics \(\sqrt{n_ i}({\hat \lambda}_{ij}-\lambda_{ij})\) are asymptotically (min \(n_ i\to \infty)\) distributed as \(N(0,2\lambda^ 2_{ij})\), independent of the \(\beta_ j\) and independent of each other \((n_ i\) being the degrees of freedom of the sample covariance matrices).
Using these results, tests are derived for (a) equality of q eigenvectors \(\beta_ 1,...,\beta_ q\) with a given set of orthonormal vectors \(\beta^ 0_ 1,...,\beta^ 0_ q\) (1\(\leq q\leq p)\), (b) neglecting p-q (out of p) principal components with relatively small variances, and (c) simultaneous sphericity of p-q common principal components. Some of the results are illustrated by a biometrical example (a three-dimensional case).

Reviewer: V.Yu.Urbakh

MSC:

62H25	Factor analysis and principal components; correspondence analysis
62E20	Asymptotic distribution theory in statistics
62H15	Hypothesis testing in multivariate analysis

Keywords:

tests for equality of eigenvectors; likelihood-ratio test; asymptotic normality; principal component analysis; asymptotic distribution; maximum likelihood estimates; simultaneous sphericity