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Testing independence and goodness-of-fit in linear models

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  • A. Sen
  • B. Sen

Abstract

We consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and predictor variables and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the predictor and the residual obtained from the parametric fit by using the Hilbert–Schmidt independence criterion (Gretton et al., 2008). The proposed method requires no user-defined regularization, is simple to compute based on only pairwise distances between points in the sample, and is consistent against all alternatives. We develop distribution theory for the proposed test statistic, under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution. We prove the consistency of the bootstrap scheme. A simulation study shows that our method has better power than its main competitors. Two real datasets are analysed to demonstrate the scope and usefulness of our method.

Suggested Citation

  • A. Sen & B. Sen, 2014. "Testing independence and goodness-of-fit in linear models," Biometrika, Biometrika Trust, vol. 101(4), pages 927-942.
    • Handle: RePEc:oup:biomet:v:101:y:2014:i:4:p:927-942.
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    File URL: http://hdl.handle.net/10.1093/biomet/asu026
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    Citations

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    Cited by:

    1. Guochang Wang & Wai Keung Li & Ke Zhu, 2018. "New HSIC-based tests for independence between two stationary multivariate time series," Papers 1804.09866, arXiv.org.
    2. Xu, Kai & Cheng, Qing, 2024. "Test of conditional independence in factor models via Hilbert–Schmidt independence criterion," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    3. Hu, Yue & Li, Haiqi & Tan, Falong, 2024. "Testing the parametric form of the conditional variance in regressions based on distance covariance," Computational Statistics & Data Analysis, Elsevier, vol. 189(C).
    4. Bodhisattva Sen & Mary Meyer, 2017. "Testing against a linear regression model using ideas from shape-restricted estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 423-448, March.
    5. Fan, Caiyun & Lu, Wenbin & Zhou, Yong, 2021. "Testing error heterogeneity in censored linear regression," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    6. Li, Shuo & Tu, Yundong, 2016. "n-consistent density estimation in semiparametric regression models," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 91-109.
    7. Patra, Rohit K. & Sen, Bodhisattva & Székely, Gábor J., 2016. "On a nonparametric notion of residual and its applications," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 208-213.
    8. Fan, Jianqing & Feng, Yang & Xia, Lucy, 2020. "A projection-based conditional dependence measure with applications to high-dimensional undirected graphical models," Journal of Econometrics, Elsevier, vol. 218(1), pages 119-139.
    9. Teran Hidalgo, Sebastian J. & Wu, Michael C. & Engel, Stephanie M. & Kosorok, Michael R., 2018. "Goodness-of-fit test for nonparametric regression models: Smoothing spline ANOVA models as example," Computational Statistics & Data Analysis, Elsevier, vol. 122(C), pages 135-155.

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