  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. 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.
