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A class of tests for a general covariance structure

Author

Listed:
  • Wakaki, Hirofumi
  • Eguchi, Shinto
  • Fujikoshi, Yasunori

Abstract

Let S be a p - p random matrix having a Wishart distribution Wp(n,n-1[Sigma]). For testing a general covariance structure [Sigma] = [Sigma]([xi]), we consider a class of test statistics Th = n inf [varrho]h(S, [Sigma]([xi])), where [varrho]h([Sigma]1, [Sigma]2) = [Sigma]j = 1ph([lambda]j) is a distance measure from [Sigma]1 to [Sigma]2, [lambda]i's are the eigenvalues of [Sigma]1[Sigma]2-1, and h is a given function with certain properties. This paper gives an asymptotic expansion of the null distribution of Th up to the order n-1. Using the general asymptotic formula, we give a condition for Th to have a Bartlett adjustment factor. Two special cases are considered in detail when [Sigma] is a linear combination or [Sigma]-1 is a linear combination of given matrices.

Suggested Citation

  • Wakaki, Hirofumi & Eguchi, Shinto & Fujikoshi, Yasunori, 1990. "A class of tests for a general covariance structure," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 313-325, February.
    • Handle: RePEc:eee:jmvana:v:32:y:1990:i:2:p:313-325
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    Citations

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    Cited by:

    1. Shimizu, Hiroaki & Wakaki, Hirofumi, 2011. "Asymptotic expansions for a class of tests for a general covariance structure under a local alternative," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1080-1089, July.
    2. Ke-Hai Yuan & Yubin Tian & Hirokazu Yanagihara, 2015. "Empirical Correction to the Likelihood Ratio Statistic for Structural Equation Modeling with Many Variables," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 379-405, June.
    3. Siotani, Minoru & Wakaki, Hirofumi, 2006. "Contributions to multivariate analysis by Professor Yasunori Fujikoshi," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 1914-1926, October.
    4. Yanagihara, Hirokazu & Tonda, Tetsuji & Matsumoto, Chieko, 2005. "The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 237-264, October.
    5. Yuan, Ke-Hai & Hayashi, Kentaro & Bentler, Peter M., 2007. "Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1262-1282, July.
    6. Marc Hallin, 2008. "On the Non Gaussian Asymptotics of the Likelihood Ratio Test Statistic for Homogeneity of Covariance," Working Papers ECARES 2008_039, ULB -- Universite Libre de Bruxelles.

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