Testing for equality of ordered eigenvectors of two multivariate normal populations

Abstract

Scatter plots for two multivariate normal data sets may suggest that the constant-density ellipsoid contours of underlying density functions have the same directions. In other words, covariance matrices of the two populations have the same eigenvectors matrix in their spectral decomposition form. In this paper, a suitable test statistic is suggested for assessing this conjecture, its asymptotic distribution under the null hypothesis is found and, based on this test statistic, a method for testing the hypothesis is provided. In the bivariate case, our method of testing is straightforward. For more than two variables a Holm–Bonferroni multiple testing procedure is used. In order to investigate the power of this simple/multiple testing procedure, some simulations are carried out. Finally, two real data sets are analysed by the use of the proposed testing method.