分位数回归中的贝叶斯变量选择
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摘要
自从Koenker&Bassett的重要论文于1978年发表以后,分位数回归已经被广泛应用到线性和非线性模型的分析中.和传统的均值模型相比,分位数模型可以更详细地描述变量的统计分布.在实际中,当存在很多个解释变量时就很有必要选择一些重要的变量以提高估计的精度.基于随机寻找变量选择方法和非对称拉普拉斯分布的混合表示,本文首先在线性模型中提出了一个简单有效的吉布斯算法进行贝叶斯变量选择.其次将其推广到具有潜在反应变量的模型中,具体的是两值和tobit分位数模型中.进一步,基于狄利克雷混合模型,本文考虑了单指标分位数模型中的贝叶斯变量选择问题,其中连接函数用截断线性样条进行构建,误差项分布由非参数狄利克雷混合模型近似,利用吉布斯抽样和MH算法进行后验推断.对于上述方法本文做了大量的数值模拟研究,结果表明提出的方法在不同的模型下均能有效的识别出真实的模型,最后将方法应用到一些实际数据中.
Since the seminal work of Koenker&Bassett[1](1978), quantile regression isgradually emerging as a comprehensive approach to the statistical analysis of linearand nonlinear response models. Compared with the mean regression, quantile re-gression can provide a more complete picture for the distribution. In practice, if thepredictor vector contains many variables, then it is necessary to select the signif-cant ones in order to improve the precision of the estimator. In this paper, based onthe stochastic search variable selection approach, we develop a simple and efcientGibbs sampling algorithm for Bayesian model selection in quantile regression basedon a location-scale mixture representation of the asymmetric laplace distribution.Also, we extended the approach in binary and tobit quantile regression. More-over, we consider variable selection in the single-index quantile regression modelbased on the stochastic search variable selection approach, where the link functionis modeled by trucated linear splines, and the distribution of the error is mod-eled nonparametrically by a Dirichlet process mixture model. Posterior inferenceis implemented using the Gibbs sampling and Metropolis-Hastings algorithm. Theabove models are illustrated using a large number of simulations. The results showthat our methods can efectively choose the true model. At last we analysis severalreal data examples.
Since the seminal work of Koenker&Bassett[1](1978), quantile regression isgradually emerging as a comprehensive approach to the statistical analysis of linearand nonlinear response models. Compared with the mean regression, quantile re-gression can provide a more complete picture for the distribution. In practice, if thepredictor vector contains many variables, then it is necessary to select the signif-cant ones in order to improve the precision of the estimator. In this paper, based onthe stochastic search variable selection approach, we develop a simple and efcientGibbs sampling algorithm for Bayesian model selection in quantile regression basedon a location-scale mixture representation of the asymmetric laplace distribution.Also, we extended the approach in binary and tobit quantile regression. More-over, we consider variable selection in the single-index quantile regression modelbased on the stochastic search variable selection approach, where the link functionis modeled by trucated linear splines, and the distribution of the error is mod-eled nonparametrically by a Dirichlet process mixture model. Posterior inferenceis implemented using the Gibbs sampling and Metropolis-Hastings algorithm. Theabove models are illustrated using a large number of simulations. The results showthat our methods can efectively choose the true model. At last we analysis severalreal data examples.
引文
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[17] Belloni, A. and Chernozhukov, V. L1quantile regression in high-dimensional sparsemodels[J]. The Annals of Statistics(2011),39:82-130.
[18] Bang, S. W. and Jhun, M. Simultaneous estimation and factor selection in quantileregression via adaptive sup-norm regularization[J]. Computational Statistics andData Analysis(2011), DOI:10.1016/j.csda.2011.01.026.
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[20] Park, T. and Casella, G. The Bayesian lasso[J]. Journal of the American StatisticalAssociation(2008),103:681-686.
[21] Hans, C. Bayesian lasso regression[J]. Biometrika(2009),96:835-845.
[22] Li, Q. and Xi, R. and Lin, N. Bayesian regularized quantile Regression[J]. BayesianAnalysis(2010),5:533-556.
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[29] Kuo, L. and Mallick, B. Variable selection for regression models[J]. Sankhya,B(1998),60:65-81.
[30] Antoniadis, A. and Gregoire, G. and McKeague, I. Bayesian estimation in single-index models[J]. Statistica Sinica(2004),14:1147-1164.
[31] Reed, C. and Dunson, D. B. and Yu, K. Bayesian quantile regression model selec-tion using stochastic search[J]. Brunel University(2010).
[32] Manski, C. F. Maximum score estimation of the stochastic utility model ofchoice[J]. Journal of Econometrics(1975),3:205-228.
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[40] Kordas, G. Smoothed binary regression quantile[J]. Journal of Applied Economet-rics(2006),21:387-407.
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