THREE STATISTICS FOR HYPOTHESIS TESTING OF INTERMEDIATE LATENT VECTOR OF COVARIANCE MATRIX AND THEIR BOOTSTRAP TESTS

Abstract

We consider the hypothesis that the α-th largest latent vector of a covariance matrix is equal to a specified vector. Anderson obtained a limiting distribution of a sample latent vector and proposed a statistic for testing the hypothesis using the result. In this paper we propose two new test statistics and derive their asymptotic expansion of the distributions under an alternative hypothesis. We compare the power of tests using three statistics under various alternative hypotheses by the asymptotic expansions. Our results show that the new statistics are superior to the statistic of Anderson in some ranges of alternative hypotheses. We discuss a bootstrap test using three statistics and test the hypothesis for real data.