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Asymptotic expansions for distributions of latent roots in multivariate analysis

Author

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  • Constantine, A. G.
  • Muirhead, R. J.

Abstract

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S1(S1 + S2)-1, where S1 is Wm(n1, [Sigma], [Omega]) and S2 is Wm(n2, [Sigma]), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix [Omega] large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n-1S, where S in Wm(n, [Sigma]), for large n, and S1S2-1, where S1 is Wm(n1, [Sigma]) and S2 is Wm(n2, [Sigma]), for large n1 + n2.

Suggested Citation

  • Constantine, A. G. & Muirhead, R. J., 1976. "Asymptotic expansions for distributions of latent roots in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 6(3), pages 369-391, September.
    • Handle: RePEc:eee:jmvana:v:6:y:1976:i:3:p:369-391
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    Cited by:

    1. Chao, John C. & Phillips, Peter C. B., 2002. "Jeffreys prior analysis of the simultaneous equations model in the case with n+1 endogenous variables," Journal of Econometrics, Elsevier, vol. 111(2), pages 251-283, December.
    2. Phillips, Peter C B, 1994. "Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models," Econometrica, Econometric Society, vol. 62(1), pages 73-93, January.
    3. Chikuse, Yasuko, 2003. "Concentrated matrix Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 375-394, May.
    4. Peter D. Hoff, 2009. "A hierarchical eigenmodel for pooled covariance estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 971-992, November.
    5. Chikuse, Yasuko, 1998. "Density Estimation on the Stiefel Manifold," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 188-206, August.
    6. Grant H. Hillier, 1987. "Joint Distribution Theory for Some Statistics Based on LIML and TSLS," Cowles Foundation Discussion Papers 840, Cowles Foundation for Research in Economics, Yale University.

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