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A characterization of spherical distributions

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  • Eaton, Morris L.

Abstract

It is shown that when the random vector X in Rn has a mean and when the conditional expectation E(u'Xv'X) = 0 for all vectors u, v [set membership, variant] Rn which satisfy u'v = 0, then the distribution of X is orthogonally invariant. A version of this characterization is also established when X does not have a mean vector.

Suggested Citation

  • Eaton, Morris L., 1986. "A characterization of spherical distributions," Journal of Multivariate Analysis, Elsevier, vol. 20(2), pages 272-276, December.
    • Handle: RePEc:eee:jmvana:v:20:y:1986:i:2:p:272-276
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    Cited by:

    1. Kai-Tai Fang & Run-Ze Li, 1997. "Some methods for generating both an NT-net and the uniform distribution on a Stiefel manifold and their applications," Computational Statistics & Data Analysis, Elsevier, vol. 24(1), pages 29-46, March.
    2. Xiangrong Yin & R. Dennis Cook, 2002. "Dimension reduction for the conditional kth moment in regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 159-175, May.
    3. Li, Lexin & Dennis Cook, R. & Nachtsheim, Christopher J., 2004. "Cluster-based estimation for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 175-193, August.
    4. Heng-Hui Lue, 2015. "An Inverse-regression Method of Dependent Variable Transformation for Dimension Reduction with Non-linear Confounding," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 760-774, September.
    5. Liu, Xuejing & Yu, Zhou & Wen, Xuerong Meggie & Paige, Robert, 2015. "On testing common indices for two multi-index models: A link-free approach," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 75-85.
    6. Heng-Hui Lue & Bing-Ran You, 2013. "High-dimensional regression analysis with treatment comparisons," Computational Statistics, Springer, vol. 28(3), pages 1299-1317, June.
    7. Zhao, Junlong & Zhao, Xiuli, 2010. "Dimension reduction using the generalized gradient direction," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 1089-1102, April.
    8. Ian Ball, 2019. "Scoring Strategic Agents," Papers 1909.01888, arXiv.org, revised Oct 2023.
    9. Wenbin Lu & Lexin Li, 2011. "Sufficient Dimension Reduction for Censored Regressions," Biometrics, The International Biometric Society, vol. 67(2), pages 513-523, June.
    10. Dong, Yuexiao & Yu, Zhou, 2012. "Dimension reduction for the conditional kth moment via central solution space," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 207-218.
    11. Maruyama Yuzo, 2003. "A robust generalized Bayes estimator improving on the James-Stein estimator for spherically symmetric distributions," Statistics & Risk Modeling, De Gruyter, vol. 21(1/2003), pages 69-78, January.
    12. Strobl Eric V. & Visweswaran Shyam, 2016. "Markov Boundary Discovery with Ridge Regularized Linear Models," Journal of Causal Inference, De Gruyter, vol. 4(1), pages 31-48, March.
    13. Heng-Hui Lue, 2010. "On principal Hessian directions for multivariate response regressions," Computational Statistics, Springer, vol. 25(4), pages 619-632, December.
    14. Gannoun, Ali & Girard, Stephane & Guinot, Christiane & Saracco, Jerome, 2004. "Sliced inverse regression in reference curves estimation," Computational Statistics & Data Analysis, Elsevier, vol. 46(1), pages 103-122, May.
    15. Albisetti, Isaia & Balabdaoui, Fadoua & Holzmann, Hajo, 2020. "Testing for spherical and elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    16. Soale, Abdul-Nasah, 2023. "Projection expectile regression for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    17. Alessandro Barbarino & Efstathia Bura, 2017. "A Unified Framework for Dimension Reduction in Forecasting," Finance and Economics Discussion Series 2017-004, Board of Governors of the Federal Reserve System (U.S.).
    18. Kariya, Takeaki & Kurata, Hiroshi, 2002. "A Maximal Extension of the Gauss-Markov Theorem and Its Nonlinear Version," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 37-55, October.
    19. Portier, François & Delyon, Bernard, 2013. "Optimal transformation: A new approach for covering the central subspace," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 84-107.
    20. Wang, Qin & Xue, Yuan, 2021. "An ensemble of inverse moment estimators for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    21. Lexin Li & Xiangrong Yin, 2008. "Sliced Inverse Regression with Regularizations," Biometrics, The International Biometric Society, vol. 64(1), pages 124-131, March.
    22. Papadatos, Nickos, 2014. "Some counterexamples concerning maximal correlation and linear regression," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 114-117.
    23. Alessandro Barbarino & Efstathia Bura, 2015. "Forecasting with Sufficient Dimension Reductions," Finance and Economics Discussion Series 2015-74, Board of Governors of the Federal Reserve System (U.S.).

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