Quantile regression with group lasso for classification

Applications of regression models for binary response are very common and models specific to these problems are widely used. Quantile regression for binary response data has recently attracted attention and regularized quantile regression methods have been proposed for high dimensional problems. When the predictors have a natural group structure, such as in the case of categorical predictors converted into dummy variables, then a group lasso penalty is used in regularized methods. In this paper, we present a Bayesian Gibbs sampling procedure to estimate the parameters of a quantile regression model under a group lasso penalty for classification problems with a binary response. Simulated and real data show a good performance of the proposed method in comparison to mean-based approaches and to quantile-based approaches which do not exploit the group structure of the predictors.     

面板数据分位数回归模型的参数估计与变量选择

本文研究了基于面板数据的分位数回归模型的变量选择问题.通过增加改进的自适应Lasso惩罚项,同时实现了固定效应面板数据的分位数回归和变量选择,得到了模型中参数的选择相合性和渐近正态性.随机模拟验证了该方法的有效性.推广了文献[14]的结论.     

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Most regression modeling is based on traditional mean regression which results in non-robust estimation results for non-normal errors. Compared to conventional mean regression, composite quantile regression (CQR) may produce more robust parameters estimation. Based on a composite asymmetric Laplace distribution (CALD), we build a Bayesian hierarchical model for the weighted CQR (WCQR). The Gibbs sampler algorithm of Bayesian WCQR is developed to implement posterior inference. Finally, the proposed method are illustrated by some simulation studies and a real data analysis.     

Bayesian joint quantile regression for mixed effects models with censoring and errors in covariates

In this paper, we discuss Bayesian joint quantile regression of mixed effects models with censored responses and errors in covariates simultaneously using Markov Chain Monte Carlo method. Under the assumption of asymmetric Laplace error distribution, we establish a Bayesian hierarchical model and derive the posterior distributions of all unknown parameters based on Gibbs sampling algorithm. Three cases including multivariate normal distribution and other two heavy-tailed distributions are considered for fitting random effects of the mixed effects models. Finally, some Monte Carlo simulations are performed and the proposed procedure is illustrated by analyzing a group of AIDS clinical data set.     

Efficient estimation and variable selection for infinite variance autoregressive models

In this paper, a self-weighted composite quantile regression estimation procedure is developed to estimate unknown parameter in an infinite variance autoregressive (IVAR) model. The proposed estimator is asymptotically normal and more efficient than a single quantile regression estimator. At the same time, the adaptive least absolute shrinkage and selection operator (LASSO) for variable selection are also suggested. We show that the adaptive LASSO based on the self-weighted composite quantile regression enjoys the oracle properties. Simulation studies and a real data example are conducted to examine the performance of the proposed approaches.     

Bayesian quantile regression model for claim count data

Quantile regression model estimates the relationship between the quantile of a response distribution and the regression parameters, and has been developed for linear models with continuous responses. In this paper, we apply Bayesian quantile regression model for the Malaysian motor insurance claim count data to study the effects of change in the estimates of regression parameters (or the rating factors) on the magnitude of the response variable (or the claim count). We also compare the results of quantile regression models from the Bayesian and frequentist approaches and the results of mean regression models from the Poisson and negative binomial. Comparison from Poisson and Bayesian quantile regression models shows that the effects of vehicle year decrease as the quantile increases, suggesting that the rating factor has lower risk for higher claim counts. On the other hand, the effects of vehicle type increase as the quantile increases, indicating that the rating factor has higher risk for higher claim counts.     

Model selection via standard error adjusted adaptive lasso

The adaptive lasso is a model selection method shown to be both consistent in variable selection and asymptotically normal in coefficient estimation. The actual variable selection performance of the adaptive lasso depends on the weight used. It turns out that the weight assignment using the OLS estimate (OLS-adaptive lasso) can result in very poor performance when collinearity of the model matrix is a concern. To achieve better variable selection results, we take into account the standard errors of the OLS estimate for weight calculation, and propose two different versions of the adaptive lasso denoted by SEA-lasso and NSEA-lasso. We show through numerical studies that when the predictors are highly correlated, SEA-lasso and NSEA-lasso can outperform OLS-adaptive lasso under a variety of linear regression settings while maintaining the same theoretical properties of the adaptive lasso.     

分位数变系数模型基于核光滑的变量选择

分位数变系数模型是一种稳健的非参数建模方法.使用变系数模型分析数据时,一个自然的问题是如何同时选择重要变量和从重要变量中识别常数效应变量.本文基于分位数方法研究具有稳健和有效性的估计和变量选择程序.利用局部光滑和自适应组变量选择方法,并对分位数损失函数施加双惩罚,我们获得了惩罚估计.通过BIC准则合适地选择调节参数,提出的变量选择方法具有oracle理论性质,并通过模拟研究和脂肪实例数据分析来说明新方法的有用性.数值结果表明,在不需要知道关于变量和误差分布的任何信息前提下,本文提出的方法能够识别不重要变量同时能区分出常数效应变量.     

Bayesian analysis of quantile regression for censored dynamic panel data

This paper develops a Bayesian approach to analyzing quantile regression models for censored dynamic panel data. We employ a likelihood-based approach using the asymmetric Laplace error distribution and introduce lagged observed responses into the conditional quantile function. We also deal with the initial conditions problem in dynamic panel data models by introducing correlated random effects into the model. For posterior inference, we propose a Gibbs sampling algorithm based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the mixture representation provides fully tractable conditional posterior densities and considerably simplifies existing estimation procedures for quantile regression models. In addition, we explain how the proposed Gibbs sampler can be utilized for the calculation of marginal likelihood and the modal estimation. Our approach is illustrated with real data on medical expenditures.     

Bayesian variable selection with sparse and correlation priors for high-dimensional data analysis

The main challenge in working with gene expression microarrays is that the sample size is small compared to the large number of variables (genes). In many studies, the main focus is on finding a small subset of the genes, which are the most important ones for differentiating between different types of cancer, for simpler and cheaper diagnostic arrays. In this paper, a sparse Bayesian variable selection method in probit model is proposed for gene selection and classification. We assign a sparse prior for regression parameters and perform variable selection by indexing the covariates of the model with a binary vector. The correlation prior for the binary vector assigned in this paper is able to distinguish models with the same size. The performance of the proposed method is demonstrated with one simulated data and two well known real data sets, and the results show that our method is comparable with other existing methods in variable selection and classification.     

Variable selection of generalized regression models based on maximum rank correlation

In this paper, we investigate the variable selection problem of the generalized regression models. To estimate the regression parameter, a procedure combining the rank correlation method and the adaptive lasso technique is developed, which is proved to have oracle properties. A modified IMO （iterative marginal optimization） algorithm which directly aims to maximize the penalized rank correlation function is proposed. The effects of the estimating procedure are illustrated by simulation studies.     

Simultaneous estimation and variable selection in median regression using Lasso-type penalty

We consider the median regression with a LASSO-type penalty term for variable selection. With the fixed number of variables in regression model, a two-stage method is proposed for simultaneous estimation and variable selection where the degree of penalty is adaptively chosen. A Bayesian information criterion type approach is proposed and used to obtain a data-driven procedure which is proved to automatically select asymptotically optimal tuning parameters. It is shown that the resultant estimator achieves the so-called oracle property. The combination of the median regression and LASSO penalty is computationally easy to implement via the standard linear programming. A random perturbation scheme can be made use of to get simple estimator of the standard error. Simulation studies are conducted to assess the finite-sample performance of the proposed method. We illustrate the methodology with a real example.     

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多元非参数分位数回归常常是难于估计的, 为了降低维数同时保持非参数估计的灵活性, 人们常常用单指标的方法模拟响应变量的条件分位数. 本文主要研究单指标分位数回归的变量选择. 以最小化平均损失估计为基础, 我们通过最小化具有SCAD惩罚项的平均损失进行变量选择和参数估计. 在正则条件下, 得到了单指标分位数回归SCAD变量选择的Oracle性质, 给出了SCAD变量选择的计算方法, 并通过模拟研究说明了本文所提方法变量选择的样本性质.     

Weighted composite quantile estimation and variable selection method for censored regression model

This paper considers the weighted composite quantile (WCQ) regression for linear model with random censoring. The adaptive penalized procedure for variable selection in this model is proposed, and the consistency, asymptotic normality and oracle property of the resulting estimators are also derived. The simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

Analyzing return asymmetry and quantiles through stochastic volatility models using asymmetric Laplace error via uniform scale mixtures

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.     

Variable selection in censored quantile regression with high dimensional data

We propose a two-step variable selection procedure for censored quantile regression with high dimensional predictors. To account for censoring data in high dimensional case, we employ effective dimension reduction and the ideas of informative subset idea. Under some regularity conditions, we show that our procedure enjoys the model selection consistency. Simulation study and real data analysis are conducted to evaluate the finite sample performance of the proposed approach.     

Bayesian Variable Selection for Median Regression

When the data has heavy tail feature or contains outliers, conventional variable selection methods based on penalized least squares or likelihood functions perform poorly. Based on Bayesian inference method, we study the Bayesian variable selection problem for median linear models. The Bayesian estimation method is proposed by using Bayesian model selection theory and Bayesian estimation method through selecting the Spike and Slab prior for regression coefficients, and the effective posterior Gibbs sampling procedure is also given. Extensive numerical simulations and Boston house price data analysis are used to illustrate the effectiveness of the proposed method.     

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在线性回归模型建模中, 回归自变量选择是一个受到广泛关注、文献众多, 具有很强的理论和实际意义的问题. 回归自变量选择子集的相合性是其中一个重要问题, 如果某种自变量选择方法选择的子集在样本量趋于无穷时是相合的, 而且预测均方误差较小, 则这种方法是可取的. 利用BIC准则可以挑选相合的自变量子集, 但是在自变量个数很多时计算量过大; 适应lasso方法具有较高计算效率, 也能找到相合的自变量子集; 本文提出一种更简单的自变量选择方法, 只需要计算两次普通线性回归: 第一次进行全集回归, 得到全集的回归系数估计, 然后利用这些回归系数估计挑选子集, 然后只要在挑选的自变量子集上再进行一次普通线性回归就得到了回归结果. 考虑如下的回归模型: 其中回归系数中非零分量下标的集合为, 设是本文方法选择的自变量子集下标集合, 是本文方法估计的回归系数(未选中的自变量对应的系数为零), 本文证明了, 在适当条件下, 其中表示的 分量下标在中的元素的组成的向量, 是误差方差, 是与 矩阵极限有关的矩阵和常数. 数值模拟结果表明本文方法具有很好的中小样本性质.     

Bayesian Sparse Group Selection

This article proposes a Bayesian approach for the sparse group selection problem in the regression model. In this problem, the variables are partitioned into different groups. It is assumed that only a small number of groups are active for explaining the response variable, and it is further assumed that within each active group only a small number of variables are active. We adopt a Bayesian hierarchical formulation, where each candidate group is associated with a binary variable indicating whether the group is active or not. Within each group, each candidate variable is also associated with a binary indicator, too. Thus, the sparse group selection problem can be solved by sampling from the posterior distribution of the two layers of indicator variables. We adopt a group-wise Gibbs sampler for posterior sampling. We demonstrate the proposed method by simulation studies as well as real examples. The simulation results show that the proposed method performs better than the sparse group Lasso in terms of selecting the active groups as well as identifying the active variables within the selected groups. Supplementary materials for this article are available online.     

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??When the data has heavy tail feature or contains outliers, conventional variable selection methods based on penalized least squares or likelihood functions perform poorly. Based on Bayesian inference method, we study the Bayesian variable selection problem for median linear models. The Bayesian estimation method is proposed by using Bayesian model selection theory and Bayesian estimation method through selecting the Spike and Slab prior for regression coefficients, and the effective posterior Gibbs sampling procedure is also given. Extensive numerical simulations and Boston house price data analysis are used to illustrate the effectiveness of the proposed method.