Abstract
In this paper, a Bayesian hierarchical model for variable selection and estimation in the context of binary quantile regression is proposed. Existing approaches to variable selection in a binary classification context are sensitive to outliers, heteroskedasticity or other anomalies of the latent response. The method proposed in this study overcomes these problems in an attractive and straightforward way. A Laplace likelihood and Laplace priors for the regression parameters are proposed and estimated with Bayesian Markov Chain Monte Carlo. The resulting model is equivalent to the frequentist lasso procedure. A conceptional result is that by doing so, the binary regression model is moved from a Gaussian to a full Laplacian framework without sacrificing much computational efficiency. In addition, an efficient Gibbs sampler to estimate the model parameters is proposed that is superior to the Metropolis algorithm that is used in previous studies on Bayesian binary quantile regression. Both the simulation studies and the real data analysis indicate that the proposed method performs well in comparison to the other methods. Moreover, as the base model is binary quantile regression, a much more detailed insight in the effects of the covariates is provided by the approach. An implementation of the lasso procedure for binary quantile regression models is available in the R-package bayesQR.
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References
Alhamzawi R, Yu K (2011) Power prior elicitation in Bayesian quantile regression. J Probab Stat. doi:10.1155/2011/874907
Alhamzawi R, Yu K (2013) Conjugate priors and variable selection for Bayesian quantile regression. Comput Stat Data Anal 64:209–219
Alhamzawi R, Yu K, Benoit DF (2012) Bayesian adaptive Lasso quantile regression. Stat Model 12:279–297
Alhamzawi R, Yu K, Pan J (2011) Prior elicitation in Bayesian quantile regression for longitudinal data. J Biom Biostat 2:115
Andrews DF, Mallows CL (1974) Scale mixtures of normal distributions. J R Stat Soc Series B Methodol 36:99–102
Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Series B Stat Methodol 63:167–241
Benoit DF, Van den Poel D (2012) Binary quantile regression: a Bayesian approach based on the asymmetric Laplace distribution. J Appl Econom 27:1174–1188
Dunson DB, Taylor JA (2005) Approximate Bayesian inference for quantiles. Nonparametr Stat 17:385–400
Florios K, Skouras S (2008) Exact computation of max weighted score estimators. J Econom 146:86–91
Geweke J (1991) Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints. In: Proceedings of the 23rd symposium on the interface, pp 571–578
Hoti F, Sillanpää MJ (2006) Bayesian map** of genotype 3 expression interactions in quantitative and qualitative traits. Heredity 97:4–18
Koenker R (2004) Quantile regression for longitudinal data. J Multivar Anal 91:74–89
Koenker R (2005) Quantile regression. Cambridge University Press, New York
Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46:33–50
Koenker RW, D’Orey V (1987) Algorithm AS 229: computing regression quantiles. Appl Stat 36:383–393
Koenker R, Machado J (1999) Goodness of fit and related inference processes for quantile regression. J Am Stat Assoc 94:1296–1310
Kordas G (2006) Smoothed binary regression quantiles. J Appl Econom 21:387–407
Kozumi H, Kobayashi G (2011) Gibbs sampling methods for Bayesian quantile regression. J Stat Comput Simul 81:1565–1578
Lancaster T, Jun SJ (2010) Bayesian quantile regression methods. J Appl Econom 25:287–307
Li Q, ** R, Lin N (2010) Bayesian regularized quantile regression. Bayesian Anal 5:1–24
Li Y, Zhu J (2008) L1-norm quantile regression. J Comput Graph Stat 17:163–185
Manski CF (1975) Maximum score estimation of the stochastic utility model of choice. J Econom 3:205–228
Manski CF (1985) Semiparametric analysis of discrete response: asymptotic properties of the maximum score estimator. J Econom 27:313–333
Michael JR, Schucany WR, Haas RW (1976) Generating random variates using transformations with multiple roots. Am Stat 30:88–90
Park T, Casella G (2008) The Bayesian lasso. J Am Stat Assoc 103:681–686
R Development Core Team (2011) R: a language and environment for statistical computing. http://www.R-project.org
Sun W, Ibrahim JG (2010) Genomewide multiple-loci map** in experimental crosses by iterative adaptive penalized regression. Genetics 185:349–359
Tibshirani R (1996) Regression Shrinkage and Selection Via the Lasso. J R Stat Soc Series B Methodol 58:267–288
Tsionas EG (2003) Bayesian quantile inference. J Stat Comput Simul 73:659–674
Wang H, Li G, Jiang G (2007) Robust regression shrinkage and consistent variable selection through the LAD-Lasso. J Bus Econ Stat 25:347–355
Wu Y, Liu Y (2009) Variable selection in quantile regression. Stat Sin 19:801–817
Yi N, Xu S (2008) Bayesian LASSO for quantitative trait loci map**. Genetics 179:1045–1055
Yu K, Lu Z, Stander J (2003) Quantile regression: applications and current research area. Statistician 52:331–350
Yu K, Moyeed RA (2001) Bayesian quantile regression. Stat Probab Lett 54:437–447
Yu K, Stander J (2007) Bayesian analysis of a Tobit quantile regression model. J Econom 137:260–276
Zheng S (2012) QBoost: predicting quantiles with boosting for regression and binary classification. Expert Syst Appl 39:1687–1697
Zou H (2006) The adaptive lasso and its oracle properties. J Am Stat Assoc 101:1418–1429
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Benoit, D.F., Alhamzawi, R. & Yu, K. Bayesian lasso binary quantile regression. Comput Stat 28, 2861–2873 (2013). https://doi.org/10.1007/s00180-013-0439-0
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DOI: https://doi.org/10.1007/s00180-013-0439-0