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Projection-averaging-based cumulative covariance and its use in goodness-of-fit testing for single-index models

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  • Xu, Kai
  • Zhou, Yeqing

Abstract

A projection-averaging-based cumulative divergence to characterize the conditional mean independence is proposed. As a natural extension of Zhou et al. (2020), the new metric has several appealing features. It ranges from zero to one, and equals zero if and only if the conditional mean independence holds. It has an elegant closed-form expression that involves no tuning parameters, making it easy to implement. The sample estimator of new metric is n-consistent under the conditional mean independence and root-n-consistent otherwise. A goodness-of-fit test for single-index models based on the variant of the proposed metric is further introduced, which generalizes the projected-based test of Escanciano (2006) to a semiparametric regression setting that allows an unspecified link function. The proposed test is consistent against any global alternatives and can detect the local alternatives distinct from the null at the parametric rate of O(n−1/2). The effectiveness of our proposals is demonstrated through simulation examples and a real application.

Suggested Citation

  • Xu, Kai & Zhou, Yeqing, 2021. "Projection-averaging-based cumulative covariance and its use in goodness-of-fit testing for single-index models," Computational Statistics & Data Analysis, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:csdana:v:164:y:2021:i:c:s0167947321001353
    DOI: 10.1016/j.csda.2021.107301
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    References listed on IDEAS

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    1. Escanciano, J. Carlos, 2006. "A Consistent Diagnostic Test For Regression Models Using Projections," Econometric Theory, Cambridge University Press, vol. 22(6), pages 1030-1051, December.
    2. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    3. Conde-Amboage, Mercedes & Sánchez-Sellero, César & González-Manteiga, Wenceslao, 2015. "A lack-of-fit test for quantile regression models with high-dimensional covariates," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 128-138.
    4. Xia, Yingcun, 2006. "Asymptotic Distributions For Two Estimators Of The Single-Index Model," Econometric Theory, Cambridge University Press, vol. 22(6), pages 1112-1137, December.
    5. Zhu, Li-Xing & Fang, Kai-Tai & Bhatti, M Ishaq, 1997. "On Estimated Projection Pursuit-Type Crámer-von Mises Statistics, ," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 1-14, October.
    6. Tingyou Zhou & Liping Zhu & Chen Xu & Runze Li, 2020. "Model-Free Forward Screening Via Cumulative Divergence," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(531), pages 1393-1405, July.
    7. Escanciano, Juan Carlos & Song, Kyungchul, 2010. "Testing single-index restrictions with a focus on average derivatives," Journal of Econometrics, Elsevier, vol. 156(2), pages 377-391, June.
    8. Wenliang Pan & Xueqin Wang & Heping Zhang & Hongtu Zhu & Jin Zhu, 2020. "Ball Covariance: A Generic Measure of Dependence in Banach Space," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 307-317, January.
    9. Xiaofeng Shao & Jingsi Zhang, 2014. "Martingale Difference Correlation and Its Use in High-Dimensional Variable Screening," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1302-1318, September.
    10. Xia, Yingcun & Härdle, Wolfgang, 2006. "Semi-parametric estimation of partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1162-1184, May.
    11. Chen, Feifei & Jiang, Qing & Feng, Zhenghui & Zhu, Lixing, 2020. "Model checks for functional linear regression models based on projected empirical processes," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    12. Liping Zhu & Kai Xu & Runze Li & Wei Zhong, 2017. "Projection correlation between two random vectors," Biometrika, Biometrika Trust, vol. 104(4), pages 829-843.
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