This paper is concerned with $2$-group linear discriminant analysis for multivariate normal populations with unknown mean vectors and unknown common covariance matrix for the case in which the sample sizes $N_1$, $N_2$ and the dimension $p$ are large. We give Studentized version of the $W$ statistic under the high-dimensional asymptotic framework A1 that $N_1$, $N_2$, 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 $(0,\infty )$. Asymptotic expansion of the distribution for the conditional probability of misclassification (CPMC) of the Studentized $W$ is derived under A1. By using this asymptotic expansion, we give the cut-off point such that the one of two CPMCs is less than the presetting value. Such the constrained discriminant rule is studied by Anderson (1973) and McLachlan (1977). Simulation result reveals that the proposed method is more accurate than McLachlan (1977)’s method for the case in which $p$ is relatively large.

We prove a Kummer-type duality theorem over certain fields without roots of unity by using algebraic tori of relative norm.

For two-way contingency tables, Tomizawa (1985) considered the quasi point-symmetry (QP) model and the marginal point-symmetry (MP) model, and gave the theorem that the point-symmetry (PS) model holds if and only if both the QP and MP models hold. Tahata and Tomizawa (2008) provided similar theorems for multi-way tables. For multi-way tables, the present paper proposes the quasi point-symmetry (QP[$f$]) model based on $f$-divergence. The QP[$f$] model includes the QP model in a special case. It also gives the theorem that the PS model holds if and only if both the QP[$f$] and MP models hold, and the theorem that the test statistic for goodness-of-fit of the PS model is asymptotically equivalent to the sum of those for the decomposed models under the PS model.

Generalized variance (GV), proposed by Wilks [8], is a one-dimensional measure of multidimensional scatter. It plays an important role in both theoretical and applied research on analyzing big data. This article examines the problem of testing equality of generalized variances of $k$ multivariate normal populations in high-dimensional and large sample settings. The conventional likelihood-ratio test statistic reveals a serious bias as dimensions increase. We present a new test statistic that eliminates this bias, and propose an asymptotic approximation-based test. The likelihood-ratio test statistic can be interpreted as an estimator of criteria related to Jensen’s inequality. Our test statistic is obtained by appropriately estimating this criteria in high-dimensional and large sample settings. In addition, our proposed test is valid not only in high dimensional settings but also in large sample settings. We also obtain the asymptotic non-null distribution of the proposed test in high-dimensional and large sample settings. Finally, we study the finite sample and dimension behavior of this test through Monte Carlo simulations.