Weighted chi-squared tests for partial common principal component subspaces

We consider tests of the null hypothesis that g covariance matrices have a partial common principal component subspace of dimension s.Our approach uses a dimensionality matrix which has its rank equal to s when the hypothesis holds. The test can then be based on a statistic computed from the eigenvalues of an estimate of this dimensionality matrix. The asymptotic distribution of this statistic is that of a linear combination of independent one-degree-offreedom chi-squared random variables. Simulation results indicate that this test yields significance levels that come closer to the nominal level than do those of a previously proposed method. The procedure is also extended to a test that g correlation matrices have a partial common principal component subspace.