Bayesian Analysis

Spatial Quantile Multiple Regression Using the Asymmetric Laplace Process

Kristian Lum and Alan E. Gelfand

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We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process (ALP) for quantile regression with spatially dependent errors. By taking advantage of a convenient conditionally Gaussian representation of the asymmetric Laplace distribution, we are able to straightforwardly incorporate spatial dependence in this process. We develop the properties of this process under several specifications, each of which induces different smoothness and covariance behavior at the extreme quantiles.

We demonstrate the advantages that may be gained by incorporating spatial dependence into this conditional quantile model by applying it to a data set of log selling prices of homes in Baton Rouge, LA, given characteristics of each house. We also introduce the asymmetric Laplace predictive process (ALPP) which accommodates large data sets, and apply it to a data set of birth weights given maternal covariates for several thousand births in North Carolina in 2000. By modeling the spatial structure in the data, we are able to show, using a check loss function, improved performance on each of the data sets for each of the quantiles at which the model was fit.

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Bayesian Anal., Volume 7, Number 2 (2012), 235-258.

First available in Project Euclid: 16 June 2012

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quantile regression conditional quantiles spatial statistics MCMC


Lum, Kristian; Gelfand, Alan E. Spatial Quantile Multiple Regression Using the Asymmetric Laplace Process. Bayesian Anal. 7 (2012), no. 2, 235--258. doi:10.1214/12-BA708.

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  • Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2004). Hierarchical modeling and analysis for spatial data. CRC Press.
  • Banerjee, S., Gelfand, A. E., Finley, A. O., and Sang, H. (2008). “Gaussian predictive process models for large spatial data sets.” Journal Of The Royal Statistical Society Series B, 70(4): 825–848.
  • Buchinsky, M. (1998). “Recent Advances in Quantile Regression Models: A Practical Guideline for Empirical Research.” The Journal of Human Resources, 33(1): 88–126.
  • Cai, Z. and Xu, X. (2008). “Nonparametric Quantile Estimations for Dynamic Smooth Coefficient Models.” Journal of the American Statistical Association, 103(484): 1595–1607.
  • Chaudhuri, P., Doksum, K., and Samarov, A. (1997). “On Average Derivative Quantile Regression.” The Annals of Statistics, 25(2): 715–744.
  • Dunson, D. B. (2007). “Empirical Bayes density regression.” Statistica Sinica, 17: 481–504.
  • Dunson, D. B., Pillai, N., and Park, J.-H. (2007). “Bayesian density regression.” Journal Of The Royal Statistical Society Series B, 69(2): 163–183.
  • Dunson, D. B. and Taylor, J. A. (2005). “Approximate Bayesian inference for quantiles.” Journal of Nonparametric Statistics, 17(3): 3850–400.
  • Finley, A. O., Sang, H., Banerjee, S., and Gelfand, A. E. (2009). “Improving the performance of predictive process modeling for large datasets.” Computational Statistics & Data Analysis, 53(8): 2873–2884.
  • Hallin, M., Lu, Z., and Yu, K. (2009). “Local linear spatial quantile regression.” Bernoulli, 15(3): 659–686.
  • Harville, D. A. (1997). Matrix algebra from a statistician’s perspective. Springer.
  • Honda, T. (2004a). “Nonparametric estimation of a conditional quantile for $\alpha$-mixing processes.” Annals of the Institute of Statistical Mathematics, 121: 113–125.
  • — (2004b). “Quantile regression in varying coefficient models.” Journal of Statistical Planning and Inferences, 121: 113–125.
  • Koenker, R. (2011). quantreg: Quantile Regression. R package version 4.71. URL
  • Koenker, R. and Bassett, G. (1978). “Regression Quantiles.” Econometrica, 46(1): 33–50.
  • Koenker, R. and Hallock, K. F. (2001). “Quantile Regression.” The Journal of Economic Perspectives, 15(4): 143–156.
  • Koenker, R., Ng, P., and Portnoy, S. (1994). “Quantile Smoothing Splines.” Biometrika, 81(4): 673–680.
  • Kottas, A. and Krnjajić, M. (2009). “Bayesian Semiparametric Modelling in Quantile Regression.” Scandinavian Journal of Statistics, 36: 297–319.
  • Kozumi, H. and Kobayashi, G. (2011). “Gibbs sampling methods for Bayesian quantile regression.” Journal of Statistical Computation and Simulation, 81(1): 1565–1578.
  • Kuzobowski, T. J. and Podgorski, K. (2000). “A Multivariate and Asymmetric Generalization of Laplace Distribution.” Computational Statistics, 15(4): 531–540.
  • Li, Q., Xi, R., and Lin, N. (2010). “Bayesian Regularized Quantile Regression.” Bayesian Analysis, 5(3): 533–556.
  • Miranda, M., Kim, D., Reiter, J., Galeano, M. O., and Maxson, P. (2009). “Environmental contributors to the achievement gap.” Neurotoxicology, 30(6): 1019–1024.
  • Reed, C. and Yu, K. (2009). “An Efficient Gibbs Sampling for Bayesian quantile regression.” Technical report, Department of Mathematical Sciences, Brunel University.
  • Reich, B. J., Bondell, H. D., and Wang, H. J. (2010). “Flexible Bayesian quantile regression for independent and clustered data.” Biostatistics, 11(2): 337–352.
  • Reich, B. J., Fuentes, M., and Dunson, D. B. (2011). “Bayesian Spatial Quantile Regression.” Journal of the American Statistical Association, 106(493): 6–20.
  • Taddy, M. and Kottas, A. (2010). “A Bayesian Nonparametric Approach to Inference for Quantile Regression.” Journal of Business and Economic Statistics, 28: 357–369.
  • Thompson, P., Cai, Y., Moyeed, R., Reeve, D., and Stander, J. (2010). “Bayesian nonparametric quantile regression using splines.” Computational Statistics & Data Analysis, 54(4): 1138 –1150.
  • Tokdar, S. T. and Kadane, J. B. (2011). “Simultaneous Linear Quantile Regression: A Semiparametric Bayesian Approach.” Bayesian Analysis, 6(4): 1–22.
  • Tsionas, E. (2003). “Bayesian Quantile Inference.” Journal of Statistical Computation and Simulation, 73: 659–674.
  • Yu, K. (2002). “Quantile regression using RJMCMC algorithm.” Computational Statistics & Data Analysis, 40: 303–315.
  • Yu, K., Lu, Z., and Stander, J. (2003). “Quantile Regression: Applications and Current Research Areas.” Journal of the Royal Statistical Society. Series D (The Statistician), 52(3): 331–350.
  • Yu, K. and Moyeed, R. A. (2001). “Bayesian quantile regression.” Statistics & Probability Letters, 54(4): 437 – 447.

See also

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  • Related item: Marco A. R. Ferreira. Comment on Article by Lum and Gelfand. Bayesian Anal., Vol. 7, Iss. 2 (2012), 271–272.
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