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The matrix angular central Gaussian distribution

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  • Chikuse, Yasuko
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    Abstract

    The Riemann space whose elements are m - k (m >= k) matrices X such that X'X = Ik is called the Stiefel manifold and denoted by Vk,m. Some distributions on Vk,m, e.g., the matrix Langevin (or von Mises-Fisher) and Bingham distributions and the uniform distribution, have been defined and discussed in the literature. In this paper, we present methods to construct new kinds of distributions on Vk,m and discuss some properties of these distributions. We investigate distributions of the "orientation" HZ = Z(Z'Z)-1/2 ([epsilon]Vk,m) of an m - k random matrix Z. The general integral form of the density of HZ reduces to a simple mathematical form, when Z has the matrix-variate central normal distribution with parameter [Sigma], an m - m positive definite matrix. We may call this distribution the matrix angular central Gaussian distribution with parameter [Sigma], denoted by the MACG ([Sigma]) distribution. The MACG distribution reduces to the angular central Gaussian distribution on the hypersphere for k = 1, which has been already known. Then, we are concerned with distributions of the orientation HY of a linear transformation Y = BZ of Z, where B is an m - m matrix such that [short parallel]B[short parallel] [not equal to] 0. Utilizing properties of these distributions, we propose a general family of distributions of Z such that HZ has the MACG ([Sigma]) distribution.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 33 (1990)
    Issue (Month): 2 (May)
    Pages: 265-274

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    Handle: RePEc:eee:jmvana:v:33:y:1990:i:2:p:265-274

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    Related research

    Keywords: Stiefel manifolds orientation of a random matrix matrix angular Gaussian distributions matrix-variate normal distributions matrix elliptically symmetric distributions;

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    Cited by:
    1. Chikuse, Yasuko, 1998. "Density Estimation on the Stiefel Manifold," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 188-206, August.
    2. Justyna Wróblewska, 2009. "Bayesian Model Selection in the Analysis of Cointegration," Central European Journal of Economic Modelling and Econometrics, CEJEME, vol. 1(1), pages 57-69, March.
    3. Strachan, Rodney W. & Inder, Brett, 2004. "Bayesian analysis of the error correction model," Journal of Econometrics, Elsevier, vol. 123(2), pages 307-325, December.
    4. Gary Koop & Simon M. Potter & Rodney W. Strachan, 2005. "Reexamining the consumption-wealth relationship: the role of model uncertainty," Staff Reports 202, Federal Reserve Bank of New York.
    5. Jupp, P. E., 2001. "Modifications of the Rayleigh and Bingham Tests for Uniformity of Directions," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 1-20, April.
    6. Hendriks, Harrie & Landsman, Zinoviy, 1998. "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 227-243, November.
    7. repec:onb:oenbwp:y::i:164:b:1 is not listed on IDEAS
    8. Sylvia Kaufmann & Johann Scharler, 2013. "Bank-Lending Standards, Loan Growth and the Business Cycle in the Euro Area," Working Papers 2013-34, Faculty of Economics and Statistics, University of Innsbruck.

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