Abstract

Estimating the structures at high or low quantiles has become an important subject and attracted

increasing attention across numerous fields. However, due to data sparsity at tails, it usually is a challenging task

to obtain reliable estimation, especially for high-dimensional data. This paper suggests a flexible parametric

structure to tails, and this enables us to conduct the estimation at quantile levels with rich observations and then

to extrapolate the fitted structures to far tails. The proposed model depends on some quantile indices and hence

is called the quantile index regression. Moreover, the composite quantile regression method is employed to

obtain non-crossing quantile estimators, and this paper further establishes their theoretical properties, including

asymptotic normality for the case with low-dimensional covariates and non-asymptotic error bounds for that

with high-dimensional covariates. Simulation studies and an empirical example are presented to illustrate the

usefulness of the new model.

Information

Preprint No.SS-2025-0069
Manuscript IDSS-2025-0069
Complete AuthorsYingying Zhang, Qianqian Zhu, Yuefeng Si, Guodong Li
Corresponding AuthorsQianqian Zhu
Emailszhu.qianqian@mail.shufe.edu.cn

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Acknowledgments

We are deeply grateful to the Co-Editor, the Associate Editor and two anonymous referees for

their valuable comments that led to the substantial improvement in the quality of this paper.

Zhang’s research was supported by the National Natural Science Foundation of China Grants

12471280 and 12101241. Zhu’s research was supported by the National Natural Science Foundation of China Grants 72373087 and 12271330. Li’s research was supported by the Hong

Kong Research Grant Council Grants 17313722 and 17309625, and the National Natural Science Foundation of China Grant 72033002.

Supplementary Materials

The online supplementary materials include the proofs of Proposition 1 and all theorems, and

additional numerical results for simulation and real data analysis.

Supplementary materials are available for download.