Pao-Lu Hsu (Xu, Bao-lu): The Grandparent of Probability and Statistics in China

The years 1910-1911 are auspicious years in Chinese mathematics with the births of Pao-Lu Hsu, Luo-Keng Hua and Shiing-Shen Chern. These three began the development of modern mathematics in China: Hsu in probability and statistics, Hua in number theory, and Chern in differential geometry. We here review some facts about the life of P.-L. Hsu which have been uncovered recently, and then discuss some of his contributions. We have drawn heavily on three papers in the 1979 Annals of Statistics (volume 7, pages 467-483) by T. W. Anderson, K. L. Chung and E. L. Lehmann, as well as an article by Jiang Ze-Han and Duan Xue-Fu in Hsu's collected papers.Comment: Published in at http://dx.doi.org/10.1214/12-STS387 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Moments of minors of Wishart matrices

For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order $m$ is populated by all $m\times m$-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many classical multivariate problems.Comment: Published in at http://dx.doi.org/10.1214/07-AOS522 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Several Colorful Inequalities

For a positive random variable X, let μα(X) be the αth moment of X. Mark Brown proves that for positive and independent random variables X, Y, F(X + Y) ≥ F(X) + F(Y), where F (X) is the ratio μ-1 over μ-2. We prove this inequality and several generalizations by a method which can be used to prove the Schwarz inequality, but which is not widely appreciated

Lattices of Graphical Gaussian Models with Symmetries

In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, model classes which place equality restrictions on concentrations or partial correlations have previously been introduced in the literature. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards-Havr\'anek model selection procedure. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other.Comment: 29 pages, 18 figures. Restructured Section 5, results unchanged; added references in Section 6; amended example in Section 6.

Matrix extensions of Liouville-Dirichlet-type integrals

AbstractThe Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x1,…,xk: xi⩾0, i=1,…,k, s⩽∑k1xi⩽T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes Ω={V1,…,Vk: Vi>0, i=1,…,k, 0⩽∑Vi⩽t}, where now each Vi is a p×p symmetric matrix and A⩾B means that A−B is positive semidefinite