When A is a $3 x 3$ positive semi-definite Hermitian matrix, Schur’s inequality and the permanental dominance conjecture are known to hold. In Sharp inequalities for the permanental dominance conjecture, we determined the possible positions of the normalized generalized matrix functions relative to the determinant and the permanent except in the case that the order of the subgroup is 2. The purpose of this paper is to determine the possible positions in the last open case.

In this paper, we attempt to select a working correlation structure for generalized estimating equations. We propose a selection criterion based on the Stein’s loss function. Our criterion consistently selects the true correlation structure when the unknown parameters are $sqrt(n)$-consistent, where $n$ is the sample size. We demonstrate the performance of the proposed methodology by a numerical study.

Homogeneous polar foliations of complex hyperbolic spaces have been classified by Berndt and Dıáz-Ramos. In this paper, we study geometry of leaves of such foliations: the minimality, the parallelism of the mean curvature vectors, and the congruency of orbits. In particular, we classify minimal leaves.