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Corrections to test statistics in principal Hessian directions

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  • Bentler, Peter M.
  • Xie, Jun

Abstract

Li's pHd method uses an asymptotic chi-squared test statistic to evaluate a hypothesized dimensionality of a reduced-dimension space in a largely nonparametric setting. This statistic is based on an assumed normal distribution of the predictors. When the distributional assumption is violated, a mixture chi-squared test proposed by Cook is theoretically more appropriate. However, both tests may not perform well with small or intermediate sized nonnormal samples. We propose two corrections to Li's statistic to enable the chi-squared approximation to be more accurate in such samples. The corrections are based on the mean and variance of the statistic of Cook's mixture distribution. The performance of Li's, Cook's, and the two new statistics are compared in some small simulation studies. Results show that one of the new tests performs about as well as Cook's, while the other performs better than the previously proposed tests.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:47:y:2000:i:4:p:381-389
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    References listed on IDEAS

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    1. Franklin Satterthwaite, 1941. "Synthesis of variance," Psychometrika, Springer;The Psychometric Society, vol. 6(5), pages 309-316, October.
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    Cited by:

    1. Charles Lindsey & Simon Sheather & Joseph McKean, 2014. "Using sliced mean variance–covariance inverse regression for classification and dimension reduction," Computational Statistics, Springer, vol. 29(3), pages 769-798, June.
    2. 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.
    3. Zhou Yu & Yuexiao Dong & Li-Xing Zhu, 2016. "Trace Pursuit: A General Framework for Model-Free Variable Selection," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 813-821, April.
    4. François Portier, 2016. "An Empirical Process View of Inverse Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 827-844, 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. Bura, E. & Yang, J., 2011. "Dimension estimation in sufficient dimension reduction: A unifying approach," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 130-142, January.
    7. 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.
    8. Artemiou, Andreas & Tian, Lipu, 2015. "Using sliced inverse mean difference for sufficient dimension reduction," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 184-190.
    9. Cook, R. Dennis & Yin, Xiangrong, 2002. "Asymptotic distributions for testing dimensionality in q-based pHd," Statistics & Probability Letters, Elsevier, vol. 58(3), pages 233-243, July.

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