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Adaptive quantile regression with precise risk bounds

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Abstract

An adaptive local smoothing method for nonparametric conditional quantile regression models is considered in this paper. Theoretical properties of the procedure are examined. The proposed method is fully adaptive in the sense that no prior information about the structure of the model is assumed. The fully adaptive feature not only allows varying bandwidths to accommodate jumps or instantaneous slope changes, but also allows the algorithm to be spatially adaptive. Under general conditions, precise risk bounds for homogeneous and heterogeneous cases of the underlying conditional quantile curves are established. An automatic selection algorithm for locally adaptive bandwidths is also given, which is applicable to higher dimensional cases. Simulation studies and data analysis confirm that the proposed methodology works well.

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Acknowledgements

The work was supported by the major research projects of Philosophy and Social Science of the Chinese Ministry of Education (Grant No. 15JZD015), National Natural Science Foundation of China (Grant No. 11271368), the major program of Beijing Philosophy and Social Science Foundation of China (Grant No. 15ZDA17), project of Ministry of Education supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130004110007), the Key Program of National Philosophy and Social Science Foundation Grant (Grant No. 13AZD064), the major project of Humanities Social Science Foundation of Ministry of Education (Grant No. 15JJD910001), Renmin University of China, the Special Developing and Guiding Fund for Building World-Class Universities (Disciplines) (Grant No. 15XNL008), China Statistical Research Project (Grant No. 2016LD03), the Fund of the Key Research Center of Humanities and Social Sciences in the general Colleges and Universities of Xinjiang Uygur Autonomous Region, General Research Fund of Hong Kong Special Administrative Region Research Grants Council General Research Fund (Grant Nos. 14300514 and 14325612), Hong Kong Special Administrative Region-Research Grants Council Collaborative Research Fund (Grant No. CityU8/CRG/12G), the Theme-Based Research Scheme of Hong Kong Special Administrative Region-Research Grants Council Theme Based Scheme (Grant No. T32-101/15-R). Part of this research was conducted while the second author was visitng Renmin University of China (RUC) through the Chang Jiang Visiting Professorship award. Research supported by the School of Statistics at RUC is gratefully acknowledged. The authors thank two anonymous referees for their constructive feedback and suggestions that improved the manuscript.

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Authors and Affiliations

  1. School of Statistics, Lanzhou University of Finance and Economics, Lanzhou, 730101, China

    MaoZai Tian

  2. School of Statistics and Information, Xinjiang University of Finance and Economics, Urumuqi, 830012, China

    MaoZai Tian

  3. Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing, 100872, China

    MaoZai Tian

  4. Department of Statistics, The Chinese University of Hong Kong, Hong Kong, China

    Ngai Hang Chan

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  1. MaoZai Tian
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Correspondence to MaoZai Tian.

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Tian, M., Chan, N.H. Adaptive quantile regression with precise risk bounds. Sci. China Math. 60, 875–896 (2017). https://doi.org/10.1007/s11425-015-0199-7

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