The Beta-MANOVA Ensemble with General Covariance
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1309.4328.pdf
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367.68 KB
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Author(s)
Dubbs, Alexander Joseph
Edelman, Alan
Date Issued
January 2014
Journal
Random Matrices: Theory and Applications
Publisher
World Scientific Pub Co Pte Lt
Citation
DUBBS, ALEXANDER, and ALAN EDELMAN. “The Beta-MANOVA Ensemble with General Covariance." Random Matrices: Theory and Applications 03, no. 01 (January 2014): 1450002.
Version
Original manuscript
Abstract
We find the joint generalized singular value distribution and largest generalized singular value distributions of the β -MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous β > 0 case for identity covariance (in eigenvalue form), and by setting the covariance to I in our model we get another version. For the diagonal covariance case, it has only been done for β = 1, 2, 4 cases (real, complex, and quaternion matrix entries). This is in a way the first second-order β-ensemble, since the sampler for the generalized singular values of the β-MANOVA with diagonal covariance calls the sampler for the eigenvalues of the β-Wishart with diagonal covariance of Forrester and Dubbs-Edelman-Koev-Venkataramana. We use a conjecture of MacDonald proven by Baker and Forrester concerning an integral of a hypergeometric function and a theorem of Kaneko concerning an integral of Jack polynomials to derive our generalized singular value distributions. In addition we use many identities from Forrester’s Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct.
Subjects
Finite random matrix theory; beta-ensembles; MANOVA
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1142/S2010326314500026
