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On model-free conditional coordinate tests for regressions

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  • Yu, Zhou
  • Zhu, Lixing
  • Wen, Xuerong Meggie

Abstract

Existing model-free tests of the conditional coordinate hypothesis in sufficient dimension reduction (Cook (1998) [3]) focused mainly on the first-order estimation methods such as the sliced inverse regression estimation (Li (1991) [14]). Such testing procedures based on quadratic inference functions are difficult to be extended to second-order sufficient dimension reduction methods such as the sliced average variance estimation (Cook and Weisberg (1991) [9]). In this article, we develop two new model-free tests of the conditional predictor hypothesis. Moreover, our proposed test statistics can be adapted to commonly used sufficient dimension reduction methods of eigendecomposition type. We derive the asymptotic null distributions of the two test statistics and conduct simulation studies to examine the performances of the tests.

Suggested Citation

  • Yu, Zhou & Zhu, Lixing & Wen, Xuerong Meggie, 2012. "On model-free conditional coordinate tests for regressions," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 61-72.
  • Handle: RePEc:eee:jmvana:v:109:y:2012:i:c:p:61-72
    DOI: 10.1016/j.jmva.2012.02.004
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    References listed on IDEAS

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    1. Bentler, Peter M. & Xie, Jun, 2000. "Corrections to test statistics in principal Hessian directions," Statistics & Probability Letters, Elsevier, vol. 47(4), pages 381-389, May.
    2. Lexin Li, 2007. "Sparse sufficient dimension reduction," Biometrika, Biometrika Trust, vol. 94(3), pages 603-613.
    3. Zhu, Yu & Zeng, Peng, 2006. "Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1638-1651, December.
    4. Li, Bing & Wang, Shaoli, 2007. "On Directional Regression for Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 997-1008, September.
    5. Xiangrong Yin, 2003. "Estimating central subspaces via inverse third moments," Biometrika, Biometrika Trust, vol. 90(1), pages 113-125, March.
    6. Prasad A. Naik & Chih-Ling Tsai, 2005. "Constrained Inverse Regression for Incorporating Prior Information," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 204-211, March.
    7. Cook, R. Dennis & Ni, Liqiang, 2005. "Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 410-428, June.
    8. Yin, Xiangrong & Li, Bing & Cook, R. Dennis, 2008. "Successive direction extraction for estimating the central subspace in a multiple-index regression," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1733-1757, September.
    9. Yongwu Shao & R. Dennis Cook & Sanford Weisberg, 2007. "Marginal tests with sliced average variance estimation," Biometrika, Biometrika Trust, vol. 94(2), pages 285-296.
    10. R. Dennis Cook & Liqiang Ni, 2006. "Using intraslice covariances for improved estimation of the central subspace in regression," Biometrika, Biometrika Trust, vol. 93(1), pages 65-74, March.
    11. Liping Zhu & Tao Wang & Lixing Zhu & Louis Ferré, 2010. "Sufficient dimension reduction through discretization-expectation estimation," Biometrika, Biometrika Trust, vol. 97(2), pages 295-304.
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    Cited by:

    1. Yu, Zhou & Dong, Yuexiao & Guo, Ranwei, 2013. "On determining the structural dimension via directional regression," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 987-992.
    2. 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.
    3. Liu, Xuejing & Huo, Lei & Wen, Xuerong Meggie & Paige, Robert, 2017. "A link-free approach for testing common indices for three or more multi-index models," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 236-245.
    4. repec:wyi:journl:002176 is not listed on IDEAS

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