Abstract
Under the common principal component model $k$ covariance matrices $\mathbf{\Sigma}_1,\cdots,\mathbf{\Sigma}_k$ are simultaneously diagonalizable, i.e., there exists an orthogonal matrix $\mathbf{\beta}$ such that $\mathbf{\beta'\Sigma_i\beta = \Lambda_i}$ is diagonal for $i = 1,\cdots, k$. In this article we give the asymptotic distribution of the maximum likelihood estimates of $\mathbf{\beta}$ and $\mathbf{\Lambda}_i$. Using these results, we derive tests for (a) equality of eigenvectors with a given set of orthonormal vectors, and (b) redundancy of $p - q$ (out of $p$) principal components. The likelihood-ratio test for simultaneous sphericity of $p - q$ principal components in $k$ populations is derived, and some of the results are illustrated by a biometrical example.
Citation
Bernard N. Flury. "Asymptotic Theory for Common Principal Component Analysis." Ann. Statist. 14 (2) 418 - 430, June, 1986. https://doi.org/10.1214/aos/1176349930
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