ON ASYMPTOTIC EXPANSION OF NULL DISTRIBUTIONS AND POWER COMPARISON OF TESTS USING THREE STATISTICS FOR HYPOTHESIS TESTING OF INTERMEDIATE LATENT VECTOR OF COVARIANCE MATRIX

Abstract

In principal component analysis, a latent vector of a covariance matrix is important for interpreting a principal component. So we consider the hypothesis testing which is an quality between the α-th largest latent vector and the specified vector. In this paper we propose two new test statistics and derive the asymptotic expansion of the distributions under the null hypothesis, though we know well the statistic given by T. W. Anderson. We obtain the power of tests using three statistics under various alternative hypotheses by a simulation study, and compare the power of tests using them. Our results show that the new statistics are superior to the statistic of Anderson in some ranges of alternative hypotheses.