IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v54y1995i1p18-31.html
   My bibliography  Save this article

Large Sample Asymptotic Theory of Tests for Uniformity on the Grassmann Manifold

Author

Listed:
  • Chikuse, Y.
  • Watson, G. S.

Abstract

The Grassmann manifold Gk,m - k consists of k-dimensional linear subspaces in Rm. To each in Gk,m - k, corresponds a unique m - m orthogonal projection matrix P idempotent of rank k. Let Pk,m - k denote the set of all such orthogonal projection matrices. We discuss distribution theory on Pk,m - k, presenting the differential form for the invariant measure and properties of the uniform distribution, and suggest a general family F(P) of non-uniform distributions. We are mainly concerned with large sample asymptotic theory of tests for uniformity on Pk,m - k. We investigate the asymptotic distribution of the standardized sample mean matrix U taken from the family F(P) under a sequence of local alternatives for large sample size n. For tests of uniformity versus the matrix Langevin distribution which belongs to the family F(P), we consider three optimal tests-the Rayleigh-style, the likelihood ratio, and the locally best invariant tests. They are discussed in relation to the statistic U, and are shown to be approximately, near uniformity, equivalent to one another. Zonal and invariant polynomials in matrix arguments are utilized in derivations.

Suggested Citation

  • Chikuse, Y. & Watson, G. S., 1995. "Large Sample Asymptotic Theory of Tests for Uniformity on the Grassmann Manifold," Journal of Multivariate Analysis, Elsevier, vol. 54(1), pages 18-31, July.
  • Handle: RePEc:eee:jmvana:v:54:y:1995:i:1:p:18-31
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(85)71043-3
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Chikuse, Yasuko, 2003. "Concentrated matrix Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 375-394, May.
    3. Chikuse, Y. & Jupp, P. E., 2004. "A test of uniformity on shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 163-176, January.
    4. Chikuse, Yasuko, 2006. "State space models on special manifolds," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1284-1294, July.
    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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:54:y:1995:i:1:p:18-31. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.