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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On disguised inverted Wishart distribution


Authors: A. K. Gupta and S. Ofori-Nyarko
Journal: Proc. Amer. Math. Soc. 123 (1995), 2557-2562
MSC: Primary 62H10
MathSciNet review: 1249879
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A \sim {W_p}(n,\Sigma )$ and $ A = ZZ'$ where Z is a lower triangular matrix with positive diagonal elements. Further, let $ B = {A^{ - 1}} = W'W$ have inverted Wishart distribution so that $ W = {Z^{ - 1}}$. In this paper we derive the distribution of $ M = W\Sigma W'$. It is also shown that $ \frac{{n - p + 1}}{{np}}T'MT \sim {F_{p,n - p + 1}}$ where $ T \sim {N_p}(0,{I_p})$ is independent of M.


References [Enhancements On Off] (What's this?)

  • [1] Walter L. Deemer and Ingram Olkin, The Jacobians of certain matrix transformations useful in multivariate analysis, Biometrika 38 (1951), 345–367. Based on lectures of P. L. Hsu at the University of North Carolina, 1947. MR 0047300 (13,855c)
  • [2] A. K. Gupta and S. Ofori-Nyarko, Improved minimax estimators of covariance and precision matrices when additional information is available on some coordinates, Department of Mathematics and Statistics, Bowling Green State University, Technical Report No. 93-11, 1993.
  • [3] W. Y. Tan and Irwin Guttman, A disguised Wishart variable and a related theorem, J. Roy. Statist. Soc. Ser. B 33 (1971), 147–152. MR 0287640 (44 #4843)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1249879-2
PII: S 0002-9939(1995)1249879-2
Keywords: Minimax estimation, risk, Jacobian, lower triangular matrix, F-distribution
Article copyright: © Copyright 1995 American Mathematical Society