Asymptotic Theory for Common Principal Component Analysis

Under the common principal component model k covariance matrices Σ₁,⋯,Σₖ are simultaneously diagonalizable, i.e., there exists an orthogonal matrix β such that β'Σᵢβ = Λᵢ is diagonal for i = 1,⋯, k. In this article we give the asymptotic distribution of the maximum likelihood estimates of β and Λᵢ. 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.