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Regularized Quantile Regression and Robust Feature Screening for Single Index Models


Wei Zhong, Liping Zhu, Runze Li and Hengjian Cui

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Regularized Quantile Regression and Robust
Feature Screening for Single Index Models
Wei Zhong, Liping Zhu, Runze Li and Hengjian Cui
Xiamen University, Shanghai University of Finance and Economics
Pennsylvania State University and Capital Normal University
Abstract:We propose both a penalized quantile regression and an independence
screening procedure to identify important covariates and to exclude unimportant
ones for a general class of ultrahigh dimensional single-index models, in which the
conditional distribution of the response depends on the covariates via a single-index
structure. We observe that the linear quantile regression yields a consistent estimator of the direction of the index parameter in the single-index model. Such an
observation dramatically reduces computational complexity in selecting important
covariates in the single-index model. We establish an oracle property for the penalized quantile regression estimator when the covariate dimension increases at an
exponential rate of the sample size. From a practical perspective, however, when
the covariate dimension is extremely large, the penalized quantile regression may
suffer from at least two drawbacks: computational expediency and algorithmic stability. To address these issues, we propose an independence screening procedure
which is robust to model misspecification, and has reliable performance when the
distribution of the response variable is heavily tailed or response realizations contain extreme values. The new independence screening procedure offers a useful
complement to the penalized quantile regression since it helps to reduce the covariate dimension from ultrahigh dimensionality to a moderate scale. Based on
the reduced model, the penalized linear quantile regression further refines selection
of important covariates at different quantile levels. We examine the finite sample performance of the newly proposed procedure by Monte Carlo simulations and
demonstrate the proposed methodology by an empirical analysis of a real data set.
Key words and phrases:Distance correlation, penalized quantile regression, singleindex models, sure screening property, ultrahigh dimensionality.

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