Bayesian Analysis

Bayesian Quantile Regression for Ordinal Models

Mohammad Arshad Rahman

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The paper introduces a Bayesian estimation method for quantile regression in univariate ordinal models. Two algorithms are presented that utilize the latent variable inferential framework of Albert and Chib (1993) and the normal-exponential mixture representation of the asymmetric Laplace distribution. Estimation utilizes Markov chain Monte Carlo simulation – either Gibbs sampling together with the Metropolis–Hastings algorithm or only Gibbs sampling. The algorithms are employed in two simulation studies and implemented in the analysis of problems in economics (educational attainment) and political economy (public opinion on extending “Bush Tax” cuts). Investigations into model comparison exemplify the practical utility of quantile ordinal models.

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Bayesian Anal., Volume 11, Number 1 (2016), 1-24.

First available in Project Euclid: 4 February 2015

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asymmetric Laplace Markov chain Monte Carlo Gibbs sampling Metropolis–Hastings educational attainment Bush Tax cuts


Rahman, Mohammad Arshad. Bayesian Quantile Regression for Ordinal Models. Bayesian Anal. 11 (2016), no. 1, 1--24. doi:10.1214/15-BA939.

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  • Albert, J. and Chib, S. (1993). “Bayesian Analysis of Binary and Polychotomous Response Data.” Journal of the American Statistical Association, 88(422): 669–679.
  • Andrews, D. F. and Mallows, C. L. (1974). “Scale Mixture of Distributions.” Journal of the Royal Statistical Society – Series B (Statistical Methodology), 36(1): 99–102.
  • Barrodale, I. and Roberts, F. D. K. (1973). “Improved Algorithm for Discrete $L_{1}$ Linear Approximation.” SIAM Journal of Numerical Analysis, 10(5): 839–848.
  • Celeux, G., Forbes, F., Robert, C. P., and Titterington, D. M. (2006). “Deviance Information Criteria for Missing Data Models.” Bayesian Analysis, 1(4): 651–674.
  • Chen, C. (2007). “A Finite Smoothing Algorithm for Quantile Regression.” Journal of Computational and Graphical Statistics, 16(1): 136–164.
  • Dagpunar, J. (1988). Principles of Random Variate Generation. Clarendon Press, Oxford.
  • — (1989). “An Easily Implemented Generalized Inverse Gaussian Generator.” Communications in Statistics – Simulation and Computation, 18(2): 703–710.
  • — (2007). Simulations and Monte Carlo: With Applications in Finance and MCMC. John Wiley & Sons Ltd., UK.
  • Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton.
  • Dantzig, G. B. and Thapa, M. N. (1997). Linear Programming 1: Introduction. Springer, New York.
  • — (2003). Linear Programming 2: Theory and Extensions. Springer, New York.
  • Goffe, W. L., Ferrier, G., and Rogers, J. (1994). “Global Optimization of Statistical Functions with Simulated Annealing.” Journal of Econometrics, 60(1–2): 65–99.
  • Greenberg, E. (2012). Introduction to Bayesian Econometrics. Cambridge University Press, New York.
  • Hong, H. G. and He, X. (2010). “Prediction of Functional Status for the Elderly Based on a New Ordinal Regression Model.” Journal of the American Statistical Association, 105(491): 930–941.
  • Hong, H. G. and Zhou, J. (2013). “A Multi-Index Model for Quantile Regression with Ordinal Data.” Journal of Applied Statistics, 40(6): 1231–1245.
  • Jeliazkov, I., Graves, J., and Kutzbach, M. (2008). “Fitting and Comparison of Models for Multivariate Ordinal Outcomes.” Advances in Econometrics: Bayesian Econometrics, 23: 115–156.
  • Karmarkar, N. (1984). “A New Polynomial Time Algorithm for Linear Programming.” Combinatorica, 4(4): 373–395.
  • Kirkpatrick, S., Gellat, C., and Vecchi, M. (1983). “Optimization by Simulated Annealing.” Science, 220(4598): 671–680.
  • Koenker, R. and Bassett, G. (1978). “Regression Quantiles.” Econometrica, 46(1): 33–50.
  • Koenker, R. and d’Orey, V. (1987). “Computing Regression Quantiles.” Journal of the Royal Statistical Society – Series C (Applied Statistics), 36(3): 383–393.
  • Kotz, S., Kozubowski, T. J., and Podgorski, K. (2001). The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance. Birkhäuser, Boston.
  • Kozumi, H. and Kobayashi, G. (2011). “Gibbs Sampling Methods for Bayesian Quantile Regression.” Journal of Statistical Computation and Simulation, 81(11): 1565–1578.
  • Madsen, K. and Nielsen, H. B. (1993). “A Finite Smoothing Algorithm for Linear $L_{1}$ Estimation.” SIAM Journal of Optimization, 3(2): 223–235.
  • Mehrotra, S. (1992). “On the Implementation of Primal–Dual Interior Point Methods.” SIAM Journal of Optimization, 2(4): 575–601.
  • Park, T. and Casella, G. (2008). “The Bayesian Lasso.” Journal of the American Statistical Association, 103(482): 681–686.
  • Portnoy, S. and Koenker, R. (1997). “The Gaussian Hare and the Laplacian Tortoise: Computability of Squared-Error versus Absolute-Error Estimators.” Statistical Science, 12(4): 279–300.
  • Rahman, M. A. (2013). “Quantile Regression using Metaheuristic Algorithms.” International Journal of Computational Economics and Econometrics, 3(3/4): 205–233.
  • Reed, C. and Yu, K. (2009). “A Partially Collapsed Gibbs Sampler for Bayesian Quantile Regression.” Computing and Mathematics Research Papers, Brunel University.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002). “Bayesian Measures of Model Complexity and Fit.” Journal of the Royal Statistical Society – Series B (Statistical Methodology), 64(4): 583–639.
  • Tsionas, E. (2003). “Bayesian Quantile Inference.” Journal of Statistical Computation and Simulation, 73(9): 659–674.
  • Yu, K. and Moyeed, R. A. (2001). “Bayesian Quantile Regression.” Statistics and Probability Letters, 54(4): 437–447.
  • Yu, K. and Zhang, J. (2005). “A Three Paramter Asymmetric Laplace Distribution and its Extensions.” Communications in Statistics – Theory and Methods, 34(9–10): 1867–1879.
  • Zhou, L. (2010). “Conditional Quantile Estimation with Ordinal Data.” Ph.D. thesis, University of South Carolina.