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Bayesian joint-quantile regression

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Abstract

Estimation of low or high conditional quantiles is called for in many applications, but commonly encountered data sparsity at the tails of distributions makes this a challenging task. We develop a Bayesian joint-quantile regression method to borrow information across tail quantiles through a linear approximation of quantile coefficients. Motivated by a working likelihood linked to the asymmetric Laplace distributions, we propose a new Bayesian estimator for high quantiles by using a delayed rejection and adaptive Metropolis and Gibbs algorithm. We demonstrate through numerical studies that the proposed estimator is generally more stable and efficient than conventional methods for estimating tail quantiles, especially at small and modest sample sizes.

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References

  • Alhamzawi R (2016) Bayesian analysis of composite quantile regression. Stat Biosci 8:358–373

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Chernozhukov V, Du S (2008) Extremal quantiles and value-at-risk. In: Durlauf SN, Blume LE (eds) The new Palgrave dictionary of economics. Palgrave Macmillan, Basingstoke

    Google Scholar 

  • Feng Y, Chen Y, He X (2015) Bayesian quantile regression with approximate likelihood. Bernoulli 21(2):832–850

    MathSciNet  MATH  Google Scholar 

  • Gardes L, Girard S (2011) Functional kernel estimators of conditional extreme quantiles. In: Ferraty F (ed) Recent advances in functional data analysis and related topics. Springer, Berlin, pp 135–140

    Google Scholar 

  • Gelman A, Roberts GO, Gilks WR (1996) Efficient Metropolis jumping rules. In: Bernardo JM, Berger JO, Dawid AP and Smith AMF (eds) Bayesian statistics 5. Oxford Univ. Press, New York, pp 599–607

    Google Scholar 

  • Geraci M, Bottai M (2007) Quantile regression for longitudinal data using the asymmetric laplace distribution. Biostatistics 8(1):140–154

    MATH  Google Scholar 

  • Green P, Mira A (2001) Delayed rejection in reversible jump Metropolis–Hastings. Biometrika 88(4):1035–1053

    MathSciNet  MATH  Google Scholar 

  • Guo J, Tian M, Zhu K (2012) New efficient and robust estimation in varying-coefficient models with heteroscedasticity. Stat Sin 22(3):1075–1101

    MathSciNet  MATH  Google Scholar 

  • Haario H, Saksman E, Tamminen J (1999) Adaptive proposal distribution for random walk metropolis algorithm. Comput Stat 14(3):375–395

    MATH  Google Scholar 

  • Haario H, Saksman E, Tamminen J (2001) An adaptive metropolis algorithm. Bernoulli 7(2):223–242

    MathSciNet  MATH  Google Scholar 

  • Haario H, Laine M, Mira A, Saksman E (2006) DRAM: efficient adaptive MCMC. Stat Comput 16(4):339–354

    MathSciNet  Google Scholar 

  • Huang HW, Chen ZX (2015) Bayesian composite quantile regression. J Stat Comput Simul 85(18):3744–3754

    MathSciNet  MATH  Google Scholar 

  • Jiang L, Wang H, Bondell HD (2013a) Interquantile shrinkage in regression models. J Comput Graph Stat 22(4):970–986

    MathSciNet  Google Scholar 

  • Jiang R, Zhou Z, Qian W, Chen Y (2013b) Two-step composite quantile regression for single-index models. Comput Stat Data Anal 64:180–191

    MathSciNet  MATH  Google Scholar 

  • Jiang R, Hu XP, Yu KM, Qian WM (2018) Composite quantile regression for massive datasets. Statistics 52(5):980–1004

    MathSciNet  MATH  Google Scholar 

  • Kai B, Li R, Zou H (2010) Local composite quantile regression smoothing: an efficient and safe alternative to local polynomial regression. J R Stat Soc B 72(1):49–69

    MathSciNet  MATH  Google Scholar 

  • Khare K, Hobert JP (2012) Geometric ergodicity of the Gibbs sampler for Bayesian quantile regression. J Multivar Anal 112:108–116

    MathSciNet  MATH  Google Scholar 

  • Koenker R (1984) A note on $L$-estimates for linear models. Stat Probab Lett 2(6):323–325

    MathSciNet  MATH  Google Scholar 

  • Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46(1):33–50

    MathSciNet  MATH  Google Scholar 

  • Kozumi H, Kobayashi G (2011) Gibbs sampling methods for Bayesian quantile regression. J Stat Comput Simul 81(11):1565–1578

    MathSciNet  MATH  Google Scholar 

  • Li Q, Xi R, Lin N (2010) Bayesian regularized quantile regression. Bayesian Anal 5:533–556

    MathSciNet  MATH  Google Scholar 

  • Mira A (2002) On Metropolis–Hastings algorithms with delayed rejection. Metron 59(3–4):231–241

    MathSciNet  MATH  Google Scholar 

  • Reich BJ, Smith LB (2013) Bayesian quantile regression for censored data. Biometrics 69:651–660

    MathSciNet  MATH  Google Scholar 

  • Reich BJ, Fuentes M, Dunson DB (2011) Bayesian spatial quantile regression. J Am Stat Assoc 106(493):6–20

    MathSciNet  MATH  Google Scholar 

  • Rodrigues T, Fan Y (2017) Regression adjustment for noncrossing Bayesian quantile regression. J Comput Graph Stat 26(2):275–284

    MathSciNet  Google Scholar 

  • Rodrigues T, Dortet-Bernadet J-L, Fan Y (2019) Pyramid quantile regression. J Comput Graph Stat 28(3):732–746

    MathSciNet  Google Scholar 

  • Sriram K, Ramamoorthi RV, Ghosh P (2012) Simultaneous Bayesian estimation of multiple quantiles with an extension to hierarchical models. Social Science Electronic Publishing, New York

    Google Scholar 

  • Sriram K, Ramamoorthi RV, Ghosh P (2016) On Bayesian quantile regression using a pseudo-joint asymmetric Laplace likelihood. Sankhyā Indian J Stat 78–A:87–104

    MathSciNet  MATH  Google Scholar 

  • Tang Y, Wang H, Liang H (2018) Composite estimation for single-index models with responses subject to detection limits. Scand J Stat 45(3):444–464

    MathSciNet  MATH  Google Scholar 

  • Tian YZ, Lian H, Tian MZ (2017) Bayesian composite quantile regression for linear mixed-effects models. Commun Stat Theory Methods 46(15):7717–7731

    MathSciNet  MATH  Google Scholar 

  • Uppala SM, Kallberg PW et al (2005) The era-40 re-analysis. Q J R Meteorol Soc 131:2961–3012

    Google Scholar 

  • Wang H, Li D (2013) Estimation of extreme conditional quantiles through power trnsformation. J Am Stat Assoc 108:1062–1074

    MATH  Google Scholar 

  • Wang K, Wang H (2016) Optimally combined estimation for tail quantile regression. Stat Sin 26(1):295–311

    MathSciNet  MATH  Google Scholar 

  • Wang H, Yang Y (2018) Bayesian quantile regression. In: Koenker R, Chernozhukov V, He X, Peng L (eds) Handbook of quantile regression, Chapman & Hall/CRC handbooks of modern statistical methods. CRC Press, Boca Raton, pp 41–54

    Google Scholar 

  • Wang H, Li D, He X (2012) Estimation of high conditional quantiles for heavy-tailed distributions. J Am Stat Assoc 107:1453–1464

    MathSciNet  MATH  Google Scholar 

  • Wu YK, Tian MZ, Tang ML (2020) General composite quantile regression: theory and methods. Commun Stat Theory Methods 49(9):2217–2236

    MathSciNet  Google Scholar 

  • Yang Y, Tokdar S (2017) Joint estimation of quantile planes over arbitrary predictor spaces. J Am Stat Assoc 112(519):1107–1120

    MathSciNet  Google Scholar 

  • Yang Y, Wang H, He X (2016) Posterior inference in Bayesian quantile regression with asymmetric Laplace likelihood. Int Stat Rev 84(3):327–344

    MathSciNet  Google Scholar 

  • Yu K, Moyeed RA (2001) Bayesian quantile regression. Stat Probab Lett 54(4):437–447

    MathSciNet  MATH  Google Scholar 

  • Yu K, Stander J (2007) Bayesian analysis of a Tobit quantile regression model. J Econom 137(1):260–276

    MathSciNet  MATH  Google Scholar 

  • Zhao Z, Xiao Z (2014) Efficient regressions via optimally combining quantile information. Econom Theory 30(6):1272–1314

    MathSciNet  MATH  Google Scholar 

  • Zou H, Yuan M (2008) Composite quantile regression and the oracle model selection theory. Ann Stat 36(3):1108–1126

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11631003, 11690012, 11726629 and 11701491), and Grant DMS-1712760 and the IR/D program from the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.

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Authors and Affiliations

  1. College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, Xinjiang Uygur Autonomous Region, People’s Republic of China

    Yingying Hu

  2. Department of Statistics, The George Washington University, Washington, DC, 20052, USA

    Huixia Judy Wang

  3. Department of Statistics, University of Michigan, Ann Arbor, MI, 48109, USA

    Xuming He

  4. School of Mathematics and Statistics and KLAS, Northeast Normal University, Changchun, 130024, Jilin Province, People’s Republic of China

    Jianhua Guo

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  1. Yingying Hu
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  2. Huixia Judy Wang
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  3. Xuming He
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  4. Jianhua Guo
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Correspondence to Yingying Hu.

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Hu, Y., Wang, H.J., He, X. et al. Bayesian joint-quantile regression. Comput Stat 36, 2033–2053 (2021). https://doi.org/10.1007/s00180-020-00998-w

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