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Bivariate quantile smoothing splines

Author

Listed:
  • X. He
  • P. Ng
  • S. Portnoy

Abstract

It has long been recognized that the mean provides an inadequate summary whereas the set of quantiles can supply a more complete description of a sample. We introduce bivariate quantile smoothing splines, which belong to the space of bilinear tensor product splines, as nonparametric estimators for the conditional quantile functions in a two‐dimensional design space. The estimators can be computed by using standard linear programming techniques and can further be used as building‐blocks for conditional quantile estimations in higher dimensions. For moderately large data sets, we recommend penalized bivariate B‐splines as approximate solutions. We use real and simulated data to illustrate the methodology proposed.

Suggested Citation

  • X. He & P. Ng & S. Portnoy, 1998. "Bivariate quantile smoothing splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(3), pages 537-550.
    • Handle: RePEc:bla:jorssb:v:60:y:1998:i:3:p:537-550
    DOI: 10.1111/1467-9868.00138
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    2. Fritsch, Markus & Haupt, Harry & Ng, Pin T., 2016. "Urban house price surfaces near a World Heritage Site: Modeling conditional price and spatial heterogeneity," Regional Science and Urban Economics, Elsevier, vol. 60(C), pages 260-275.
    3. Charlier, Isabelle & Paindaveine, Davy & Saracco, Jérôme, 2015. "Conditional quantile estimation based on optimal quantization: From theory to practice," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 20-39.
    4. Yufeng Liu & Yichao Wu, 2011. "Simultaneous multiple non-crossing quantile regression estimation using kernel constraints," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 415-437.
    5. Racine, Jeffrey S. & Li, Kevin, 2017. "Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach," Journal of Econometrics, Elsevier, vol. 201(1), pages 72-94.
    6. Sungil Kwak & Stephen C. Smith, 2013. "Regional Agricultural Endowments and Shifts of Poverty Trap Equilibria: Evidence from Ethiopian Panel Data," Journal of Development Studies, Taylor & Francis Journals, vol. 49(7), pages 955-975, July.
    7. Zongwu Cai & Qi Li, 2013. "Some Recent Develop- ments on Nonparametric Econometrics," Working Papers 2013-10-14, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    8. repec:wyi:journl:002094 is not listed on IDEAS
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    10. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
    11. Cheng Peng & Stanislav Uryasev, 2023. "Factor Model of Mixtures," Papers 2301.13843, arXiv.org, revised Mar 2023.
    12. Victor Chernozhukov & Kaspar Wuthrich & Yinchu Zhu, 2019. "Distributional conformal prediction," Papers 1909.07889, arXiv.org, revised Aug 2021.
    13. Zou, Hui & Yuan, Ming, 2008. "Regularized simultaneous model selection in multiple quantiles regression," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5296-5304, August.
    14. Cai, Zongwu & Xiao, Zhijie, 2012. "Semiparametric quantile regression estimation in dynamic models with partially varying coefficients," Journal of Econometrics, Elsevier, vol. 167(2), pages 413-425.
    15. Simone Fiori & Andrea Vitali, 2019. "Statistical Modeling of Trivariate Static Systems: Isotonic Models," Data, MDPI, vol. 4(1), pages 1-29, January.
    16. Bousebata, Meryem & Enjolras, Geoffroy & Girard, Stéphane, 2023. "Extreme partial least-squares," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    17. Wu, Chaojiang & Yu, Yan, 2014. "Partially linear modeling of conditional quantiles using penalized splines," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 170-187.
    18. Cai, Zongwu & Chen, Linna & Fang, Ying, 2018. "A semiparametric quantile panel data model with an application to estimating the growth effect of FDI," Journal of Econometrics, Elsevier, vol. 206(2), pages 531-553.
    19. Marcus Alexander & Matthew Harding & Carlos Lamarche, 2011. "Quantifying the impact of economic crises on infant mortality in advanced economies," Applied Economics, Taylor & Francis Journals, vol. 43(24), pages 3313-3323.
    20. Craig, Steven G. & Ng, Pin T., 2001. "Using Quantile Smoothing Splines to Identify Employment Subcenters in a Multicentric Urban Area," Journal of Urban Economics, Elsevier, vol. 49(1), pages 100-120, January.
    21. Mammen, Enno & Van Keilegom, Ingrid & Yu, Kyusang, 2013. "Expansion for Moments of Regression Quantiles with Applications to Nonparametric Testing," LIDAM Discussion Papers ISBA 2013027, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    22. He X. & Zhu L-X., 2003. "A Lack-of-Fit Test for Quantile Regression," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 1013-1022, January.
    23. Monica Pratesi & M. Ranalli & Nicola Salvati, 2009. "Nonparametric -quantile regression using penalised splines," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(3), pages 287-304.

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