Gibbs sampling for mixture quantile regression based on asymmetric Laplace distribution

In this paper, we consider the finite mixture of quantile regression model from a Bayesian perspective by assuming the errors have the asymmetric Laplace distribution (ALD), and develop the Gibbs sampling algorithm to estimate various quantile conditional on covariate in different groups using the Normal-Exponential representation of the ALD. We conduct several simulations under different error distributions to demonstrate the performance of the algorithm, and finally apply it to analyse a real data set, finding that the procedure has good performance.     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

Bayesian quantile regression for longitudinal data models

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained.     

Bayesian Elastic Net Tobit Quantile Regression

In this article, the problem of parameter estimation and variable selection in the Tobit quantile regression model is considered. A Tobit quantile regression with the elastic net penalty from a Bayesian perspective is proposed. Independent gamma priors are put on the l₁ norm penalty parameters. A novel aspect of the Bayesian elastic net Tobit quantile regression is to treat the hyperparameters of the gamma priors as unknowns and let the data estimate them along with other parameters. A Bayesian Tobit quantile regression with the adaptive elastic net penalty is also proposed. The Gibbs sampling computational technique is adapted to simulate the parameters from the posterior distributions. The proposed methods are demonstrated by both simulated and real data examples.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

A non-iterative posterior sampling algorithm for linear quantile regression model

In this article, a non-iterative posterior sampling algorithm for linear quantile regression model based on the asymmetric Laplace distribution is proposed. The algorithm combines the inverse Bayes formulae, sampling/importance resampling, and the expectation maximization algorithm to obtain independently and identically distributed samples approximately from the observed posterior distribution, which eliminates the convergence problems in the iterative Gibbs sampling and overcomes the difficulty in evaluating the standard deviance in the EM algorithm. The numeric results in simulations and application to the classical Engel data show that the non-iterative sampling algorithm is more effective than the Gibbs sampling and EM algorithm.     

基于Gibbs抽样算法的面板数据分位回归方法

 文章讨论了含有随机效应的面板数据模型,利用非对称Laplace分布与分位回归之间的关系,文章建立了一种贝叶斯分层分位回归模型。通过对非对称Laplace分布的分解,文章给出了Gibbs抽样算法下模型参数的点估计及区间估计,模拟结果显示,在处理含随机效应的面板数据模型中,特别是在误差非正态的情况下,本文的方法优于传统的均值模型方法。文章最后利用新方法对我国各地区经济与就业面板数据进行了实证研究,得到了有利于宏观调控的有用信息。     

基于贝叶斯方法的比例数据分位数推断及其应用

为了尝试使用贝叶斯方法研究比例数据的分位数回归统计推断问题,首先基于Tobit模型给出了分位数回归建模方法,然后通过选取合适的先验分布得到了贝叶斯层次模型,进而给出了各参数的后验分布并用于Gibbs抽样。数值模拟分析验证了所提出的贝叶斯推断方法对于比例数据分析的有效性。最后,将贝叶斯方法应用于美国加州海洛因吸毒数据,在不同的分位数水平下揭示了吸毒频率的影响因素。     

Variable selection in quantile regression via Gibbs sampling

Due to computational challenges and non-availability of conjugate prior distributions, Bayesian variable selection in quantile regression models is often a difficult task. In this paper, we address these two issues for quantile regression models. In particular, we develop an informative stochastic search variable selection (ISSVS) for quantile regression models that introduces an informative prior distribution. We adopt prior structures which incorporate historical data into the current data by quantifying them with a suitable prior distribution on the model parameters. This allows ISSVS to search more efficiently in the model space and choose the more likely models. In addition, a Gibbs sampler is derived to facilitate the computation of the posterior probabilities. A major advantage of ISSVS is that it avoids instability in the posterior estimates for the Gibbs sampler as well as convergence problems that may arise from choosing vague priors. Finally, the proposed methods are illustrated with both simulation and real data.     

Bayesian composite Tobit quantile regression

Composite quantile regression models have been shown to be effective techniques in improving the prediction accuracy [H. Zou and M. Yuan, Composite quantile regression and the oracle model selection theory, Ann. Statist. 36 (2008), pp. 1108–1126; J. Bradic, J. Fan, and W. Wang, Penalized composite quasi-likelihood for ultrahighdimensional variable selection, J. R. Stat. Soc. Ser. B 73 (2011), pp. 325–349; Z. Zhao and Z. Xiao, Efficient regressions via optimally combining quantile information, Econometric Theory 30(06) (2014), pp. 1272–1314]. This paper studies composite Tobit quantile regression (TQReg) from a Bayesian perspective. A simple and efficient MCMC-based computation method is derived for posterior inference using a mixture of an exponential and a scaled normal distribution of the skewed Laplace distribution. The approach is illustrated via simulation studies and a real data set. Results show that combine information across different quantiles can provide a useful method in efficient statistical estimation. This is the first work to discuss composite TQReg from a Bayesian perspective.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Bayesian quantile regression for parametric nonlinear mixed effects models

We propose quantile regression (QR) in the Bayesian framework for a class of nonlinear mixed effects models with a known, parametric model form for longitudinal data. Estimation of the regression quantiles is based on a likelihood-based approach using the asymmetric Laplace density. Posterior computations are carried out via Gibbs sampling and the adaptive rejection Metropolis algorithm. To assess the performance of the Bayesian QR estimator, we compare it with the mean regression estimator using real and simulated data. Results show that the Bayesian QR estimator provides a fuller examination of the shape of the conditional distribution of the response variable. Our approach is proposed for parametric nonlinear mixed effects models, and therefore may not be generalized to models without a given model form.     

Do maternal health problems influence child's worrying status? Evidence from the British Cohort Study

Conventional methods apply symmetric prior distributions such as a normal distribution or a Laplace distribution for regression coefficients, which may be suitable for median regression and exhibit no robustness to outliers. This work develops a quantile regression on linear panel data model without heterogeneity from a Bayesian point of view, i.e. upon a location-scale mixture representation of the asymmetric Laplace error distribution, and provides how the posterior distribution is summarized using Markov chain Monte Carlo methods. Applying this approach to the 1970 British Cohort Study (BCS) data, it finds that a different maternal health problem has different influence on child's worrying status at different quantiles. In addition, applying stochastic search variable selection for maternal health problems to the 1970 BCS data, it finds that maternal nervous breakdown, among the 25 maternal health problems, contributes most to influence the child's worrying status.     

A gradient search maximization algorithm for the asymmetric Laplace likelihood

The asymmetric Laplace likelihood naturally arises in the estimation of conditional quantiles of a response variable given covariates. The estimation of its parameters entails unconstrained maximization of a concave and non-differentiable function over the real space. In this note, we describe a maximization algorithm based on the gradient of the log-likelihood that generates a finite sequence of parameter values along which the likelihood increases. The algorithm can be applied to the estimation of mixed-effects quantile regression, Laplace regression with censored data, and other models based on Laplace likelihood. In a simulation study and in a number of real-data applications, the proposed algorithm has shown notable computational speed.     

Bayesian bridge quantile regression

Regularization methods for simultaneous variable selection and coefficient estimation have been shown to be effective in quantile regression in improving the prediction accuracy. In this article, we propose the Bayesian bridge for variable selection and coefficient estimation in quantile regression. A simple and efficient Gibbs sampling algorithm was developed for posterior inference using a scale mixture of uniform representation of the Bayesian bridge prior. This is the first work to discuss regularized quantile regression with the bridge penalty. Both simulated and real data examples show that the proposed method often outperforms quantile regression without regularization, lasso quantile regression, and Bayesian lasso quantile regression.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.     

基于逆跳MCMC的贝叶斯分位自回归模型研究

考虑到传统信息理论方法确定模型存在不足,在贝叶斯理论框架下提出了基于逆跳马尔可夫链蒙特卡罗法确定分位自回归模型阶次的方法。在时间序列服从非对称Laplace分布的条件下,设计了马尔可夫链蒙特卡罗数值计算程序,得到了不同分位数下模型参数的贝叶斯估计值。实证研究表明：基于逆跳马尔可夫链蒙特卡罗法的贝叶斯分位自回归模型能有效地揭示滞后变量对响应变量的位置、尺度和形状的影响。     

Bayesian variable selection and estimation in maximum entropy quantile regression

Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis.     

Bayesian quantile regression for single-index models

Using an asymmetric Laplace distribution, which provides a mechanism for Bayesian inference of quantile regression models, we develop a fully Bayesian approach to fitting single-index models in conditional quantile regression. In this work, we use a Gaussian process prior for the unknown nonparametric link function and a Laplace distribution on the index vector, with the latter motivated by the recent popularity of the Bayesian lasso idea. We design a Markov chain Monte Carlo algorithm for posterior inference. Careful consideration of the singularity of the kernel matrix, and tractability of some of the full conditional distributions leads to a partially collapsed approach where the nonparametric link function is integrated out in some of the sampling steps. Our simulations demonstrate the superior performance of the Bayesian method versus the frequentist approach. The method is further illustrated by an application to the hurricane data.