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Distance Metrics for Measuring Joint Dependence with Application to Causal Inference

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  • Shubhadeep Chakraborty
  • Xianyang Zhang

Abstract

Many statistical applications require the quantification of joint dependence among more than two random vectors. In this work, we generalize the notion of distance covariance to quantify joint dependence among d≥2 random vectors. We introduce the high-order distance covariance to measure the so-called Lancaster interaction dependence. The joint distance covariance is then defined as a linear combination of pairwise distance covariances and their higher-order counterparts which together completely characterize mutual independence. We further introduce some related concepts including the distance cumulant, distance characteristic function, and rank-based distance covariance. Empirical estimators are constructed based on certain Euclidean distances between sample elements. We study the large-sample properties of the estimators and propose a bootstrap procedure to approximate their sampling distributions. The asymptotic validity of the bootstrap procedure is justified under both the null and alternative hypotheses. The new metrics are employed to perform model selection in causal inference, which is based on the joint independence testing of the residuals from the fitted structural equation models. The effectiveness of the method is illustrated via both simulated and real datasets. Supplementary materials for this article are available online.

Suggested Citation

  • Shubhadeep Chakraborty & Xianyang Zhang, 2019. "Distance Metrics for Measuring Joint Dependence with Application to Causal Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(528), pages 1638-1650, October.
  • Handle: RePEc:taf:jnlasa:v:114:y:2019:i:528:p:1638-1650
    DOI: 10.1080/01621459.2018.1513364
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    Cited by:

    1. Emanuele Borgonovo & Elmar Plischke & Giovanni Rabitti, 2022. "Interactions and computer experiments," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1274-1303, September.
    2. Yaxue Yan & Weijuan Liang & Banban Wang & Xiaoling Zhang, 2023. "Spillover effect among independent carbon markets: evidence from China’s carbon markets," Economic Change and Restructuring, Springer, vol. 56(5), pages 3065-3093, October.
    3. Roy, Angshuman & Ghosh, Anil K., 2020. "Some tests of independence based on maximum mean discrepancy and ranks of nearest neighbors," Statistics & Probability Letters, Elsevier, vol. 164(C).
    4. Kalinke, Florian & Szabo, Zoltan, 2024. "The minimax rate of HSIC estimation for translation-invariant kernel," LSE Research Online Documents on Economics 122819, London School of Economics and Political Science, LSE Library.
    5. Beaulieu Guillaume Boglioni & de Micheaux Pierre Lafaye & Ouimet Frédéric, 2021. "Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin," Dependence Modeling, De Gruyter, vol. 9(1), pages 424-438, January.
    6. Zhu, Hanbing & Li, Rui & Zhang, Riquan & Lian, Heng, 2020. "Nonlinear functional canonical correlation analysis via distance covariance," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    7. Marc Hallin & Simos Meintanis & Klaus Nordhausen, 2024. "Consistent Distribution–Free Affine–Invariant Tests for the Validity of Independent Component Models," Working Papers ECARES 2024-04, ULB -- Universite Libre de Bruxelles.
    8. Meintanis, Simos G. & Hušková, Marie & Hlávka, Zdeněk, 2022. "Fourier-type tests of mutual independence between functional time series," Journal of Multivariate Analysis, Elsevier, vol. 189(C).

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