Statistics and Computing

, Volume 25, Issue 2, pp 189–202 | Cite as

Functional data analysis of generalized regression quantiles

  • Mengmeng Guo
  • Lan Zhou
  • Jianhua Z. HuangEmail author
  • Wolfgang Karl Härdle


Generalized regression quantiles, including the conditional quantiles and expectiles as special cases, are useful alternatives to the conditional means for characterizing a conditional distribution, especially when the interest lies in the tails. We develop a functional data analysis approach to jointly estimate a family of generalized regression quantiles. Our approach assumes that the generalized regression quantiles share some common features that can be summarized by a small number of principal component functions. The principal component functions are modeled as splines and are estimated by minimizing a penalized asymmetric loss measure. An iterative least asymmetrically weighted squares algorithm is developed for computation. While separate estimation of individual generalized regression quantiles usually suffers from large variability due to lack of sufficient data, by borrowing strength across data sets, our joint estimation approach significantly improves the estimation efficiency, which is demonstrated in a simulation study. The proposed method is applied to data from 159 weather stations in China to obtain the generalized quantile curves of the volatility of the temperature at these stations.


Asymmetric loss function Functional data analysis Generalized quantiles Iteratively reweighted least squares Principal component analysis Penalized splines 


  1. Anastasiadou, Z., López-Cabrera, B.: Statistical modelling of temperature risk. SFB discussing paper 2012-029. Stat. Sci. (2012, submitted) Google Scholar
  2. Benth, F., Benth, J., Koekebakker, S.: Putting a price on temperature. Scand. J. Stat. 34(4), 746–767 (2007) zbMATHMathSciNetGoogle Scholar
  3. Breckling, J., Chambers, R.: M-Quantiles. Biometrika 74(4), 761–772 (1988) CrossRefMathSciNetGoogle Scholar
  4. Campbell, S., Diebold, F.: Weather forecasting for weather derivatives. J. Am. Stat. Assoc. 469, 6–16 (2005) CrossRefMathSciNetGoogle Scholar
  5. Eilers, P., Marx, B.: Flexible smoothing with B-splines and penalties. J. Am. Stat. Assoc. 11, 89–121 (1996) zbMATHMathSciNetGoogle Scholar
  6. Guo, M., Härdle, W.: Simultaneous confidence bands for expectile functions. AStA Adv. Stat. Anal. 96, 517–542 (2012) CrossRefMathSciNetGoogle Scholar
  7. Härdle, W., López-Cabrera, B.: Implied market price of weather risk. Appl. Math. Finance 5, 1–37 (2011) Google Scholar
  8. Härdle, W., Song, S.: Confidence bands in quantile regression. Econom. Theory 26(4), 1180–1200 (2010) CrossRefzbMATHGoogle Scholar
  9. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, New York (2009) CrossRefzbMATHGoogle Scholar
  10. Hunter, D., Lange, K.: Quantile regression via an MM algorithm. J. Comput. Graph. Stat. 9(1), 60–77 (2000) MathSciNetGoogle Scholar
  11. James, G., Hastie, T., Sugar, C.: Principal component models for sparse functional data. Biometrika 87, 587–602 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  12. Jones, M.: Expectiles and M-quantiles are quantiles. Stat. Probab. Lett. 20, 149–153 (1994) CrossRefzbMATHGoogle Scholar
  13. Koenker, R.: Quantile Regression. Econometric Society Monographs. Cambridge University Press, Cambridge (2005) CrossRefzbMATHGoogle Scholar
  14. Koenker, R., Bassett, G.W.: Regression quantiles. Econometrica 46, 33–50 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  15. Koenker, R., Machado, J.: Goodness of fit and related inference processes for quantile regression. J. Am. Stat. Assoc. 94, 1296–1310 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  16. Nelder, J., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965) CrossRefzbMATHGoogle Scholar
  17. Newey, W.K., Powell, J.L.: Asymmetric least squares estimation and testing. Econometrica 55, 819–847 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  18. Odening, M., Berg, E., Turvey, C.: Management of climate risk in agriculture. Agric. Financ. Rev. 68(1), 83–97 (2008). Special issue CrossRefGoogle Scholar
  19. Ramsay, J., Silverman, B.: Functional Data Analysis, 2nd edn. Springer, New York (2005) Google Scholar
  20. Ruppert, D., Wand, M., Carroll, R.: Semiparametric Regression. Cambridge University Press, Cambridge (2003) CrossRefzbMATHGoogle Scholar
  21. Schnabel, S., Eilers, P.: An analysis of life expectancy and economic production using expectile frontier zones. Demogr. Res. 21, 109–134 (2009a) CrossRefGoogle Scholar
  22. Schnabel, S., Eilers, P.: Optimal expectile smoothing. Comput. Stat. Data Anal. 53, 4168–4177 (2009b) CrossRefzbMATHMathSciNetGoogle Scholar
  23. Taylor, J.: Estimating value at risk and expected shortfall using expectiles. J. Financ. Econom. 6, 231–252 (2008) CrossRefGoogle Scholar
  24. Yao, Q., Tong, H.: Asymmetric least squares regression estimation: a nonparametric approach. J. Nonparametr. Stat. 6(2–3), 273–292 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  25. Zhou, L., Huang, J.Z., Carroll, R.: Joint modelling of paired sparse functional data using principle components. Biometrika 95(3), 601–619 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  26. Zhou, L., Huang, J.Z., Martinez, J.G., Maity, A., Baladandayuthapani, V., Carroll, R.J.: Reduced rank mixed effects models for spatially correlated hierarchical functional data. J. Am. Stat. Assoc. 105, 390–400 (2010) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Mengmeng Guo
    • 1
  • Lan Zhou
    • 2
  • Jianhua Z. Huang
    • 2
    Email author
  • Wolfgang Karl Härdle
    • 3
    • 4
  1. 1.Research Institute of Economics and ManagementSouthwestern University of Finance and EconomicsChengduChina
  2. 2.Department of StatisticsTexas A&M UniversityCollege StationUSA
  3. 3.Chair of Statistics and Center for Applied Statistics and EconomicsHumboldt-Universität zu BerlinBerlinGermany
  4. 4.Business School Quantitative FinanceSingapore Management UniversitySingaporeSingapore

Personalised recommendations