Bayesian Analysis

Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density

Karthik Sriram, R.V. Ramamoorthi, and Pulak Ghosh

Full-text: Open access

Abstract

We explore an asymptotic justification for the widely used and empirically verified approach of assuming an asymmetric Laplace distribution (ALD) for the response in Bayesian Quantile Regression. Based on empirical findings, Yu and Moyeed (2001) argued that the use of ALD is satisfactory even if it is not the true underlying distribution. We provide a justification to this claim by establishing posterior consistency and deriving the rate of convergence under the ALD misspecification. Related literature on misspecified models focuses mostly on i.i.d. models which in the regression context amounts to considering i.i.d. random covariates with i.i.d. errors. We study the behavior of the posterior for the misspecified ALD model with independent but non identically distributed response in the presence of non-random covariates. Exploiting the specific form of ALD helps us derive conditions that are more intuitive and easily seen to be satisfied by a wide range of potential true underlying probability distributions for the response. Through simulations, we demonstrate our result and also find that the robustness of the posterior that holds for ALD fails for a Gaussian formulation, thus providing further support for the use of ALD models in quantile regression.

Article information

Source
Bayesian Anal., Volume 8, Number 2 (2013), 479-504.

Dates
First available in Project Euclid: 24 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ba/1369407561

Digital Object Identifier
doi:10.1214/13-BA817

Mathematical Reviews number (MathSciNet)
MR3066950

Zentralblatt MATH identifier
1329.62308

Keywords
Asymmetric Laplace density Bayesian Quantile Regression Misspecified models Posterior consistency

Citation

Sriram, Karthik; Ramamoorthi, R.V.; Ghosh, Pulak. Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density. Bayesian Anal. 8 (2013), no. 2, 479--504. doi:10.1214/13-BA817. https://projecteuclid.org/euclid.ba/1369407561


Export citation

References

  • Amewou-Atisso, M., Ghosal, S., Ghosh, J. K., and Ramamoorthi, R. V. (2003). “Posterior consistency for semi-parametric regression problems.” Bernoulli, 9(2): 291–312.
  • Berk, R. H. (1966). “Limiting behavior of posterior distributions when the model is incorrect.” The Annals of Mathematical Statistics, 37(1): 51–58.
  • Bunke, O. and Milhaud, X. (1998). “Asymptotic behavior of Bayes estimates under possibly incorrect models.” The Annals of Statistics, 26(2): 617–644.
  • Ghosal, S. and van der Vaart, A. W. (2007). “Convergence rates of posterior distributions for noniid observations.” The Annals of Statistics, 35(1): 192–223.
  • Hu, Y., Gramacy, R. B., and Lian, H. (2012). “Bayesian quantile regression for single-index models.” Statistics and Computing, 1–18.
  • Kleijn, B. J. K. and van der Vaart, A. W. (2006). “Misspecification in infinite-dimensional Bayesian statistics.” The Annals of Statistics, 34(2): 837–877.
  • Kleijn, B. J. K. and van der Vaart, A. W. (2012). “The Bernstein-Von-Mises theorem under misspecification.” Electronic Journal of Statistics, 6: 354–381.
  • Koenker, R. (2005). Quantile Regression (Econometric Society Monographs). Cambridge University Press.
  • Koenker, R. and Bassett Jr, G. (1978). “Regression quantiles.” Econometrica, 46(1): 33–50.
  • Kozumi, H. and Kobayashi, G. (2011). “Gibbs sampling methods for Bayesian quantile regression.” Journal of statistical computation and simulation, 81(11): 1565–1578.
  • Shalizi, C. R. (2009). “Dynamics of Bayesian updating with dependent data and misspecified models.” Electronic Journal of Statistics, 3: 1039–1074.
  • Yu, K. and Moyeed, R. A. (2001). “Bayesian quantile regression.” Statistics and Probability Letters, 54(4): 437–447.
  • Yu, K., van Kerm, P., and Zhang, J. (2005). “Bayesian quantile regression: an application to the wage distribution in 1990s Britain.” Sankhyā, 67(2): 359–377.
  • Yu, K. and Zhang, J. (2005). “A three-parameter asymmetric Laplace distribution and its extension.” Communications in Statistics-Theory and Methods, 34: 1867–1879.
  • Yue, Y. R. and Rue, H. (2011). “Bayesian inference for additive mixed quantile regression models.” Computational Statistics and Data Analysis, 55: 84–96.