Statistics and Computing

, Volume 23, Issue 4, pp 437–454 | Cite as

Bayesian quantile regression for single-index models



Using an asymmetric Laplace distribution, which provides a mechanism for Bayesian inference of quantile regression models, we develop a fully Bayesian approach to fitting single-index models in conditional quantile regression. In this work, we use a Gaussian process prior for the unknown nonparametric link function and a Laplace distribution on the index vector, with the latter motivated by the recent popularity of the Bayesian lasso idea. We design a Markov chain Monte Carlo algorithm for posterior inference. Careful consideration of the singularity of the kernel matrix, and tractability of some of the full conditional distributions leads to a partially collapsed approach where the nonparametric link function is integrated out in some of the sampling steps. Our simulations demonstrate the superior performance of the Bayesian method versus the frequentist approach. The method is further illustrated by an application to the hurricane data.


Gaussian process prior Markov chain Monte Carlo Quantile regression Single-index models 



We thank the Associate Editor and three anonymous referees for their helpful comments that have led to a significant improvement of the manuscript. Robert Gramacy would like to thank the Kemper Family Foundation for their support. The research of Heng Lian is supported by Singapore Ministry of Education Tier 1 Grant.


  1. Antoniadis, A., Grégoire, G., McKeague, I.: Bayesian estimation in single-index models. Stat. Sin. 14(4), 1147–1164 (2004) MATHGoogle Scholar
  2. Barndorff-Nielsen, O., Shephard, N.: Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc., Ser. B, Stat. Methodol. 63(2), 167–241 (2001) MathSciNetMATHCrossRefGoogle Scholar
  3. Bondell, H., Reich, B., Wang, H.: Noncrossing quantile regression curve estimation. Biometrika 97(4), 825–838 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. Carvalho, C., Polson, N., Scott, J.: The horseshoe estimator for sparse signals. Biometrika 97(2), 465–480 (2010) MathSciNetMATHCrossRefGoogle Scholar
  5. Choi, T., Shi, J., Wang, B.: A Gaussian process regression approach to a single-index model. J. Nonparametr. Stat. 23(1), 21–36 (2011) MathSciNetMATHCrossRefGoogle Scholar
  6. George, E., Foster, D.: Calibration and empirical Bayes variable selection. Biometrika 87(4), 731–747 (2000) MathSciNetMATHCrossRefGoogle Scholar
  7. Geraci, M., Bottai, M.: Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8(1), 140–154 (2007) MATHCrossRefGoogle Scholar
  8. Gramacy, R., Lian, H.: Gaussian process single-index models as emulators for computer experiments. Technometrics 54(1), 30–41 (2012) MathSciNetCrossRefGoogle Scholar
  9. Griffin, J.E., Brown, P.J.: Inference with normal-gamma prior distributions in regression problems. Bayesian Anal. 5(1), 171–188 (2010) MathSciNetCrossRefGoogle Scholar
  10. Hans, C.: Bayesian lasso regression. Biometrika 96(4), 835–845 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. Härdle, W., Hall, P., Ichimura, H.: Optimal smoothing in single-index models. Ann. Stat. 157–178 (1993) Google Scholar
  12. Jagger, T., Elsner, J.: Modeling tropical cyclone intensity with quantile regression. Int. J. Climatol. 29(10), 1351–1361 (2009) CrossRefGoogle Scholar
  13. Kottas, A., Krnjajić, M.: Bayesian semiparametric modelling in quantile regression. Scand. J. Stat. 36(2), 297–319 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. Kozumi, H., Kobayashi, G.: Gibbs sampling methods for Bayesian quantile regression. J. Stat. Comput. Simul. (2011, to appear) Google Scholar
  15. Lancaster, T., Jun, S.J.: Bayesian quantile regression methods. J. Appl. Econom. 25(2), 287–307 (2010) MathSciNetCrossRefGoogle Scholar
  16. Li, Q., Lin, N.: The Bayesian elastic net. Bayesian Anal. 5(1), 151–170 (2010) MathSciNetCrossRefGoogle Scholar
  17. Liang, H., Liu, X., Li, R., Tsai, C.: Estimation and testing for partially linear single-index models. Ann. Stat. 38(6), 3811–3836 (2010) MathSciNetMATHCrossRefGoogle Scholar
  18. Liu, J.: Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer, Berlin (2008) MATHGoogle Scholar
  19. Luo, Y., Lian, H., Tian, M.: Bayesian quantile regression for longitudinal data model. J. Stat. Comput. Simul. (2011). doi: 10.1080/00949655.2011.590488 Google Scholar
  20. MacKay, D.: Introduction to Gaussian Processes. NATO ASI Series F Computer and Systems Sciences, vol. 168, pp. 133–166 (1998) Google Scholar
  21. Park, T., Casella, G.: The Bayesian lasso. J. Am. Stat. Assoc. 103(482), 681–686 (2008) MathSciNetMATHCrossRefGoogle Scholar
  22. Reich, B., Bondell, H., Wang, H.: Flexible Bayesian quantile regression for independent and clustered data. Biostatistics 11(2), 337–352 (2010) CrossRefGoogle Scholar
  23. Tokdar, S., Kadane, J.: Simultaneous linear quantile regression: a semiparametric Bayesian approach. Bayesian Anal. 7(1), 51–72 (2012) MathSciNetGoogle Scholar
  24. van Dyk, D., Park, T.: Partially collapsed Gibbs samplers. J. Am. Stat. Assoc. 103(482), 790–796 (2008) MATHCrossRefGoogle Scholar
  25. Wang, H.: Bayesian estimation and variable selection for single index models. Comput. Stat. Data Anal. 53(7), 2617–2627 (2009) MATHCrossRefGoogle Scholar
  26. Wang, J., Xue, L., Zhu, L., Chong, Y.: Estimation for a partial-linear single-index model. Ann. Stat. 38(1), 246–274 (2010) MathSciNetMATHGoogle Scholar
  27. Wu, T., Yu, K., Yu, Y.: Single-index quantile regression. J. Multivar. Anal. 101(7), 1607–1621 (2010) MathSciNetMATHCrossRefGoogle Scholar
  28. Xia, Y., Li, W., Tong, H., Zhang, D.: A goodness-of-fit test for single-index models. Stat. Sin. 14, 1–39 (2004) MathSciNetMATHGoogle Scholar
  29. Yu, K., Moyeed, R.: Bayesian quantile regression. Stat. Probab. Lett. 54(4), 437–447 (2001) MathSciNetMATHCrossRefGoogle Scholar
  30. Yu, Y., Ruppert, D.: Penalized spline estimation for partially linear single-index models. J. Am. Stat. Assoc. 97(460), 1042–1054 (2002) MathSciNetMATHCrossRefGoogle Scholar
  31. Zellner, A.: On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In: Goel, P.K., Zellner, A. (eds.) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. North-Holland/Elsevier, Amsterdam (1986) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, SPMSNanyang Technological UniversitySingaporeSingapore
  2. 2.Booth School of BusinessUniversity of ChicagoChicagoUSA

Personalised recommendations