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Oracle Estimation of a Change Point in High-Dimensional Quantile Regression

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  • Sokbae Lee
  • Yuan Liao
  • Myung Hwan Seo
  • Youngki Shin

Abstract

In this article, we consider a high-dimensional quantile regression model where the sparsity structure may differ between two sub-populations. We develop ℓ1-penalized estimators of both regression coefficients and the threshold parameter. Our penalized estimators not only select covariates but also discriminate between a model with homogenous sparsity and a model with a change point. As a result, it is not necessary to know or pretest whether the change point is present, or where it occurs. Our estimator of the change point achieves an oracle property in the sense that its asymptotic distribution is the same as if the unknown active sets of regression coefficients were known. Importantly, we establish this oracle property without a perfect covariate selection, thereby avoiding the need for the minimum level condition on the signals of active covariates. Dealing with high-dimensional quantile regression with an unknown change point calls for a new proof technique since the quantile loss function is nonsmooth and furthermore the corresponding objective function is nonconvex with respect to the change point. The technique developed in this article is applicable to a general M-estimation framework with a change point, which may be of independent interest. The proposed methods are then illustrated via Monte Carlo experiments and an application to tipping in the dynamics of racial segregation. Supplementary materials for this article are available online.

Suggested Citation

  • Sokbae Lee & Yuan Liao & Myung Hwan Seo & Youngki Shin, 2018. "Oracle Estimation of a Change Point in High-Dimensional Quantile Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1184-1194, July.
  • Handle: RePEc:taf:jnlasa:v:113:y:2018:i:523:p:1184-1194
    DOI: 10.1080/01621459.2017.1319840
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    References listed on IDEAS

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    1. Jelena Bradic & Jianqing Fan & Weiwei Wang, 2011. "Penalized composite quasi‐likelihood for ultrahigh dimensional variable selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(3), pages 325-349, June.
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    10. Alexandre Belloni & Victor Chernozhukov, 2009. "L1-Penalized Quantile Regression in High-Dimensional Sparse Models," Papers 0904.2931, arXiv.org, revised Sep 2019.
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    1. Sokbae Lee & Yuan Liao & Myung Hwan Seo & Youngki Shin, 2018. "Factor-Driven Two-Regime Regression," Papers 1810.11109, arXiv.org, revised Jul 2019.

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