Asymptotic properties of the first principal component and equality tests of covariance matrices in high-dimension, low-sample-size context

A common feature of high-dimensional data is that the data dimension is high, however, the sample size is relatively low. We call such data HDLSS data. In this paper, we study asymptotic properties of the first principal component in the HDLSS context and apply them to equality tests of covariance matrices for high-dimensional data sets. We consider HDLSS asymptotic theories as the dimension grows for both the cases when the sample size is fixed and the sample size goes to infinity. We introduce an eigenvalue estimator by the noise-reduction methodology and provide asymptotic distributions of the largest eigenvalue in the HDLSS context. We construct a confidence interval of the first contribution ratio and give a one-sample test. We give asymptotic properties both for the first PC direction and PC score as well. We apply the findings to equality tests of two covariance matrices in the HDLSS context. We provide numerical results and discussions about the performances both on the estimates of the first PC and the equality tests of two covariance matrices.