92 research outputs found
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 is populated by all -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
Reversal of the Lyapunov, Hölder, and Minkowski inequalities and other extensions of the Kantorovich inequality
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
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
Integral Expressions for Tail Probabilities for Multinomial and Negative Multinomial Distributions
1 online resource (PDF, 29 pages
A Multivariate Test of the Null Hypothesis That Two Populations Do Not Differ in the Direction of Their Mean Vectors
1 online resource (PDF, 6 pages
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