In this paper, we derive the asymptotic distributions of the characteristic roots in multivariate linear models when the dimension $p$ and the sample size $n$ are large. The results are given for the case that the population characteristic roots have multiplicities greater than unity, and their orders are $\mathrm{O}(np)$ or $\mathrm{O}(n)$. Next, similar results are given for the asymptotic distributions of the canonical correlations when one of the dimensions and the sample size are large, assuming that the order of the population canonical correlations is $\mathrm{O}(\sqrt{p})$ or $\mathrm{O}(1)$.

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 consider a new two-sample test for high-dimensional data under the strongly spiked eigenvalue (SSE) model. We consider the distance-based two-sample test under the SSE model. We introduce the noise-reduction (NR) methodology and apply that to the two-sample test. Finally, we give simulation studies and demonstrate the new test procedure by using microarray data sets.

For a class of positive homogeneously presented cancellative monoids whose heights are greater than or equal to 2, we will present several explicit calculations of the skew growth functions for them. By the inversion formula, the spherical growth functions for them can be determined. For most of them, the direct calculations are not known. The datum of certain lemmas for proving the cancellativity of the monoids are indispensable to the calculations of the skew growth functions. By improving the technique to show the lemmas, we succeed in the calculations.

This paper is concerned with the problem of classifying an observation vector into one of two populations $\mathit{\Pi}_{1} : N_{p}(\mu_{1},\Sigma)$ and $\mathit{\Pi}_{2} : N_{p}(\mu_{2},\Sigma)$. Anderson (1973, Ann. Statist.) provided an asymptotic expansion of the distribution for a Studentized linear discriminant function, and proposed a cut-off point in the linear discriminant rule to control one of the two misclassification probabilities. However, as dimension $p$ becomes larger, the precision worsens, which is checked by simulation. Therefore, in this paper we derive an asymptotic expansion of the distribution of a linear discriminant function up to the order $p^{-1}$ as $N_1$, $N_2$, and $p$ tend to infinity together under the condition that $p/(N_{1}+N_{2}-2)$ converges to a constant in $(0, 1)$, and $N_{1}/N_{2}$ converges to a constant in $(0, \infty)$, where $N_i$ means the size of sample drown from $\mathit{\Pi}_i(i=1, 2)$. Using the expansion, we provide a cut-off point. A small-scale simulation revealed that our proposed cut-off point has good accuracy.

We study biharmonic homogeneous hypersurfaces in Riemannian symmetric spaces associated to the exceptional Lie groups $\mathrm{E}_6$ and $\mathrm{G}_2$ as well as real, complex and quaternion Grassmannian manifolds.