Principal components in the nonnormal case: The test of equality of Q roots

The limiting distribution of the likelihood ratio statistic Wq for testing the hypothesis of equality of q characteristic roots of a covariance matrix is studied in the case of nonnormal populations. It is shown, both theoretically and empirically, that the limiting distribution of Wq is not robust to departures from normality and that Wq cannot be used for nonnormal populations with long tails. For the class of elliptical populations the limiting distribution of Wq can be expressed in close form. A corrected test statistic, Wq*, is proposed that can be used in principal component analysis when sampling from elliptical populations.