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Independence tests in the presence of measurement errors: An invariance law

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  • Fan, Jinlin
  • Zhang, Yaowu
  • Zhu, Liping

Abstract

In many scientific areas the observations are collected with measurement errors. We are interested in measuring and testing independence between random vectors which are subject to measurement errors. We modify the weight functions in the classic distance covariance such that, the modified distance covariance between the random vectors of primary interest is the same as its classic version between the surrogate random vectors, which is referred to as the invariance law in the present context. The presence of measurement errors may substantially weaken the degree of nonlinear dependence. An immediate issue arises: The classic distance correlation between the surrogate vectors cannot reach one even if the two random vectors of primary interest are exactly linearly dependent. To address this issue, we propose to estimate the distance variance using repeated measurements. We study the asymptotic properties of the modified distance correlation thoroughly. In addition, we demonstrate its finite-sample performance through extensive simulations and a real-world application.

Suggested Citation

  • Fan, Jinlin & Zhang, Yaowu & Zhu, Liping, 2022. "Independence tests in the presence of measurement errors: An invariance law," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:jmvana:v:188:y:2022:i:c:s0047259x21000968
    DOI: 10.1016/j.jmva.2021.104818
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    References listed on IDEAS

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    1. Alexandre Belloni & Victor Chernozhukov & Abhishek Kaul, 2017. "Confidence bands for coefficients in high dimensional linear models with error-in-variables," CeMMAP working papers 22/17, Institute for Fiscal Studies.
    2. Chi-Lun Cheng & Chih-Ling Tsai, 2004. "The Invariance of Some Score Tests in the Linear Model With Classical Measurement Error," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 805-809, January.
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    4. Li, Mengyan & Li, Runze & Ma, Yanyuan, 2021. "Inference in high dimensional linear measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
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    6. Wang, Yining & Wang, Jialei & Balakrishnan, Sivaraman & Singh, Aarti, 2019. "Rate optimal estimation and confidence intervals for high-dimensional regression with missing covariates," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
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    8. Li-Pang Chen, 2021. "Feature screening based on distance correlation for ultrahigh-dimensional censored data with covariate measurement error," Computational Statistics, Springer, vol. 36(2), pages 857-884, June.
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