Skip to main content
SpringerLink
Log in

Bayesian regularisation in geoadditive expectile regression

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Regression modelling beyond the mean of the response has found a lot of attention in the last years. Expectile regression is a special and computationally convenient case of this type of models where expectiles offer a quantile-like characterisation of the complete distribution and include the mean as a special case. In the frequentist framework, expectile regression could be combined with covariate effects of quite different forms and in particular nonlinear and spatial effects. We propose Bayesian expectile regression based on the asymmetric normal distribution as an auxiliary likelihood to allow for the additional inclusion of Bayesian regularisation priors for covariates with linear effects. Proposal densities based on iteratively weighted least squares updates for the resulting Markov chain Monte Carlo simulation algorithm are developed and evaluated in both simulations and an application. A special focus of the simulations lies on the evaluation of coverage properties of the Bayesian credible bands and the quantification of the detrimental effect arising from the misspecification of the auxiliary likelihood.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Brezger, A., Lang, S.: Generalized additive regression based on Bayesian P-splines. Comput. Stat. Data Anal. 50, 967–991 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Chernozhukov, V., Hong, H.: An mcmc approach to classical estimation. J. Econom. 115(2), 293–346 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • De Rossi, G., Harvey, A.: Quantiles, expectiles and splines. J. Econom. 152(2), 179–185 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Efron, B.: Regression percentiles using asymmetric squared error loss. Stat. Sin. 1, 93–125 (1991)

    MathSciNet  MATH  Google Scholar 

  • Eilers, P.H.C., Marx, B.D.: Flexible smoothing using B-splines and penalized likelihood. Stat. Sci. 11, 89–121 (1996)

    Article  MATH  Google Scholar 

  • Fahrmeir, L., Kneib, T., Konrath, S.: Bayesian regularisation in structured additive regression: a unifying perspective on shrinkage, smoothing and predictor selection. Stat. Comput. 20(2), 203–219 (2010)

    Article  MathSciNet  Google Scholar 

  • Fahrmeir, L., Kneib, T., Lang, S.: Penalized structured additive regression for space-time data: a Bayesian perspective. Stat. Sin. 14, 731–761 (2004)

    MathSciNet  MATH  Google Scholar 

  • Fahrmeir, L., Kneib, T., Lang, S., Marx, B.: Regression—Models, Methods and Applications. Springer, Berlin (2013)

    MATH  Google Scholar 

  • Farooq, M., Steinwart, I.: An svm-like approach for expectile regression. arXiv preprint arXiv:1507.03887 (2015)

  • Fenske, N., Kneib, T., Hothorn, T.: Identifying risk factors for severe childhood malnutrition by boosting additive quantile regression. J. Am. Stat. Assoc. 106(494), 494–510 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Gerlach, R., Chen, C.W.S.: Bayesian expected shortfall forecasting incorporating the intraday range. J. Financ. Econom. 14(1), 128–158 (2016)

    Google Scholar 

  • Kandala, N.B., Lang, S., Klasen, S., Fahrmeir, L.: Semiparametric analysis of the socio-demographic and spatial determinants of undernutrition in two african countries. University of Munich, Munich (2001)

  • Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33–50 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Koenker, R., Ng, P., Portnoy, S.: Quantile smoothing splines. Biometrika 81, 673–680 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Kozumi, H., Kobayashi, G.: Gibbs sampling methods for bayesian quantile regression. J. Stat. Comput. Simul. 81, 1565–1578 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Lum, C., Gelfand, A.: Spatial quantile multiple regression using the asymmetric laplace process. Bayesian Anal. 7(2), 235–258 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Majumdar, A., Paul, D.: Zero expectile processes and bayesian spatial regression. J. Comput. Graph. Stat. 25(3), 727–747 (2016)

    Article  MathSciNet  Google Scholar 

  • Newey, W.K., Powell, J.L.: Asymmetric least squares estimation and testing. Econometrica 55, 819–847 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Reed, C., Yu, K.: A partially collapsed gibbs sampler for bayesian quantile regression. Department of Mathematical Sciences, Brunel University, Technical Report (2009)

  • Reich, B.J., Bondell, H.D., Wang, H.: Flexible bayesian quantile regression for independent and clustered data. Biostatistics 11, 337–352 (2010)

    Article  Google Scholar 

  • R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2013). http://www.R-project.org

  • Schnabel, S.K., Eilers, P.: Optimal expectile smoothing. Comput. Stat. Data Anal. 53, 4168–4177 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Schnabel, S.K., Eilers, P.: Expectile sheets for joint estimation of expectile curves. Technical Report (2012)

  • Schulze Waltrup, L., Sobotka, F., Kneib, T., Kauermann, G.: Expectile and quantile regression—David and Goliath? Stat. Modell. 15(5), 433–456 (2015)

    Article  MathSciNet  Google Scholar 

  • Sobotka, F., Kauermann, G., Schulze Waltrup, L., Kneib, T.: On confidence intervals for semiparametric expectile regression. Stat. Comput. 23(2), 135–148 (2013a)

    Article  MathSciNet  MATH  Google Scholar 

  • Sobotka, F., Kneib, T.: Geoadditive expectile regression. Comput. Stat. Data Anal. 56, 755–767 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Sobotka, F., Marra, G., Radice, R., Kneib, T.: Estimating the relationship between women’s education and fertility in Botswana by using an instrumental variable approach to semiparametric expectile regression. J. R. Stat. Soc. 62(1), 25–45 (2013b)

    Article  MathSciNet  Google Scholar 

  • Sobotka, F., Schnabel, S., Schulze Waltrup, L.: Expectreg: Expectile and Quantile Regression. R Package Version 0.39. Georg August University Goettingen, Goettingen (2014)

  • Sriram, K., Ramamoorthi, R., Ghosh, P.: Indian Institute of Management, B.: Posterior consistency of bayesian quantile regression under a mis-specified likelihood based on asymmetric laplace density. Bayesian. Analysis 8, 479–504 (2013)

    MATH  Google Scholar 

  • Taylor, J.W.: Estimating value at risk and expected shortfall using expectiles. J. Financ. Econom. 6, 231–252 (2008)

    Article  Google Scholar 

  • Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58, 267–288 (1994)

    MathSciNet  MATH  Google Scholar 

  • Waldmann, E., Kneib, T., Yue, Y.R., Lang, S., Flexeder, C.: Bayesian semiparametric additive quantile regression. Stat. Modell. 13(3), 223–252 (2013)

    Article  MathSciNet  Google Scholar 

  • Yang, Y., Zou, H.: Nonparametric multiple expectile regression via er-boost. J. Stat. Comput. Simul. 84(1), 84–95 (2015)

    Article  MathSciNet  Google Scholar 

  • Yao, Q., Tong, H.: Asymmetric least squares regression estimation: a nonparametric approach. J. Nonparamet. Stat. 6(2–3), 273–292 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Yu, K., Moyeed, R.A.: Bayesian quantile regression. Stat. Probab. Lett. 54, 437–447 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Yue, Y., Rue, H.: Bayesian inference for additive mixed quantile regression models. Comput. Stat. Data Anal. 55, 84–96 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Ziegel, J.F.: Coherence and elicitability. Math. Financ. doi: 10.1111/mafi.12080 (2014)

Download references

Author information

Authors and Affiliations

  1. Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

    Elisabeth Waldmann

  2. Carl-von-Ossietzky Universität Oldenburg, Oldenburg, Germany

    Fabian Sobotka

  3. Georg-August-Universität Göttingen, Göttingen, Germany

    Thomas Kneib

Authors
  1. Elisabeth Waldmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabian Sobotka
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Kneib
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elisabeth Waldmann.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 2288 KB)

Supplementary material 2 (zip 17 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Waldmann, E., Sobotka, F. & Kneib, T. Bayesian regularisation in geoadditive expectile regression. Stat Comput 27, 1539–1553 (2017). https://doi.org/10.1007/s11222-016-9703-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-016-9703-9

Keywords

Search

Navigation