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Perturbation Inequalities and Confidence Sets for Functions of a Scatter Matrix

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  • Dümbgen, Lutz

Abstract

Let[Sigma]be an unknown covariance matrix. Perturbation (in)equalities are derived for various scale-invariant functionals of[Sigma]such as correlations (including partial, multiple and canonical correlations) or angles between eigenspaces. These results show that a particular confidence set for[Sigma]is canonical if one is interested in simultaneous confidence bounds for these functionals. The confidence set is based on the ratio of the extreme eigenvalues of[Sigma]-1S, whereSis an estimator for[Sigma]. Asymptotic considerations for the classical Wishart model show that the resulting confidence bounds are substantially smaller than those obtained by inverting likelihood ratio tests.

Suggested Citation

  • Dümbgen, Lutz, 1998. "Perturbation Inequalities and Confidence Sets for Functions of a Scatter Matrix," Journal of Multivariate Analysis, Elsevier, vol. 65(1), pages 19-35, April.
    • Handle: RePEc:eee:jmvana:v:65:y:1998:i:1:p:19-35
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    References listed on IDEAS

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    1. Eaton, Morris L., 1976. "A maximization problem and its application to canonical correlation," Journal of Multivariate Analysis, Elsevier, vol. 6(3), pages 422-425, September.
    2. Dumbgen, L., 1995. "Likelihood Ratio Tests for Principal Components," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 245-258, February.
    3. Rao, C. Radhakrishna, 1979. "Separation theorems for singular values of matrices and their applications in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 9(3), pages 362-377, September.
    4. Jeyaratnam, S., 1992. "Confidence intervals for the correlation coefficient," Statistics & Probability Letters, Elsevier, vol. 15(5), pages 389-393, December.
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