Asymptotic expansions of the null distribution of the MANOVA test statistics including the likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai tests are obtained when both the sample size and the dimension tend to infinity with assuming the ratio of the dimension and the sample size tends to a positive constant smaller than one. Cornish-Fisher expansions of the upper percent points are also obtained. In order to study the accuracy of the approximation formulas, some numerical experiments are done, with comparing to the classical expansions when only the sample size tends to infinity.

We give a lower bound on the Walsh figure of merit (WAFOM), which estimates the integration error for quasi-Monte Carlo (QMC) integration by a point set called a digital net. The logarithm of this lower bound is optimal up to a constant multiple, because the existence of point sets attaining the order was proved in K. Suzuki, "An explicit construction of point sets with large minimum Dick weight," to appear in J. Complexity.

We give a topological description of orbit spaces and orbits of some flat Lorentzian G-manifolds.

In the structural equation modeling, unknown parameters of a covariance matrix are derived by minimizing the discrepancy between a sample covariance matrix and a covariance matrix having a specified structure. When a sample covariance matrix is a near singular matrix, Yuan and Chan (2008) proposed the estimation method to use an adjusted sample covariance matrix instead of the sample covariance matrix in the discrepancy function. The adjusted sample covariance matrix is defined by adding a scalar matrix with a shrinkage parameter to the existing sample covariance matrix. They used a constant value as the shrinkage parameter, which was chosen based solely on the sample size and the number of dimensions of the observation, and not on the data itself. However, selecting the shrinkage parameter from the data may lead to a greater improvement in prediction compared to the use of a constant shrinkage parameter. Hence, we propose an information criterion for selecting the shrinkage parameter, and attempt to select the shrinkage parameter by an information criterion minimization method. The proposed information criterion is based on the discrepancy function measured by the normal theory maximum likelihood. Using the Monte Carlo method, we demonstrate that the proposed criterion works well in the sense that the prediction accuracy of an estimated covariance matrix is improved.

Cassels proved that projectively equivalent integral quadratic forms are commensurable. In this note, an elementary proof of the converse of this theorem, for indefinite forms, is given. This was proved in "On integral quadratic forms having commensurable groups of automorphisms," Hiroshima Math. J. 43, 371–411 (2013) for forms of Sylvester signature +++. . .+- or ---. . .-+ (hyperbolic forms) and it was left there, as an open problem, for non-hyperbolic indefinite forms of any Sylvester signature.