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.     

Multi-step quantile regression tree

Quantile regression (QR) proposed by Koenker and Bassett [Regression quantiles, Econometrica 46(1) (1978), pp. 33–50] is a statistical technique that estimates conditional quantiles. It has been widely studied and applied to economics. Meinshausen [Quantile regression forests, J. Mach. Learn. Res. 7 (2006), pp. 983–999] proposed quantile regression forests (QRF), a non-parametric way based on random forest. QRF performs well in terms of prediction accuracy, but it struggles with noisy data sets. This motivates us to propose a multi-step QR tree method using GUIDE (Generalized, Unbiased, Interaction Detection and Estimation) made by Loh [Regression trees with unbiased variable selection and interaction detection, Statist. Sinica 12 (2002), pp. 361–386]. Our simulation study shows that the multi-step QR tree performs better than a single tree or QRF especially when it deals with data sets having many irrelevant variables.     

Bayesian composite quantile regression

One advantage of quantile regression, relative to the ordinary least-square (OLS) regression, is that the quantile regression estimates are more robust against outliers and non-normal errors in the response measurements. However, the relative efficiency of the quantile regression estimator with respect to the OLS estimator can be arbitrarily small. To overcome this problem, composite quantile regression methods have been proposed in the literature which are resistant to heavy-tailed errors or outliers in the response and at the same time are more efficient than the traditional single quantile-based quantile regression method. This paper studies the composite quantile regression from a Bayesian perspective. The advantage of the Bayesian hierarchical framework is that the weight of each component in the composite model can be treated as open parameter and automatically estimated through Markov chain Monte Carlo sampling procedure. Moreover, the lasso regularization can be naturally incorporated into the model to perform variable selection. The performance of the proposed method over the single quantile-based method was demonstrated via extensive simulations and real data analysis.     

A weighted quantile regression for left-truncated and right-censored data

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. In this article, it is demonstrated that the estimating equation of Zhou [A weighted quantile regression for randomly truncated data, Comput. Stat. Data Anal. 55 (2011), pp. 554–566.] can be extended to analyse left-truncated and right-censored data. We evaluate the finite sample performance of the proposed estimators through simulation studies. The proposed estimator β?(q) is applied to the Veteran's Administration lung cancer data reported by Prentice [Exponential survival with censoring and explanatory variables, Biometrika 60 (1973), pp. 279–288].     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.     

A note on the efficiency of composite quantile regression

Composite quantile regression (CQR) is motivated by the desire to have an estimator for linear regression models that avoids the breakdown of the least-squares estimator when the error variance is infinite, while having high relative efficiency even when the least-squares estimator is fully efficient. Here, we study two weighting schemes to further improve the efficiency of CQR, motivated by Jiang et al. [Oracle model selection for nonlinear models based on weighted composite quantile regression. Statist Sin. 2012;22:1479–1506]. In theory the two weighting schemes are asymptotically equivalent to each other and always result in more efficient estimators compared with CQR. Although the first weighting scheme is hard to implement, it sheds light on in what situations the improvement is expected to be large. A main contribution is to theoretically and empirically identify that standard CQR has good performance compared with weighted CQR only when the error density is logistic or close to logistic in shape, which was not noted in the literature.     

Bayesian Tobit quantile regression using g-prior distribution with ridge parameter

A Bayesian approach is proposed for coefficient estimation in the Tobit quantile regression model. The proposed approach is based on placing a g-prior distribution depends on the quantile level on the regression coefficients. The prior is generalized by introducing a ridge parameter to address important challenges that may arise with censored data, such as multicollinearity and overfitting problems. Then, a stochastic search variable selection approach is proposed for Tobit quantile regression model based on g-prior. An expression for the hyperparameter g is proposed to calibrate the modified g-prior with a ridge parameter to the corresponding g-prior. Some possible extensions of the proposed approach are discussed, including the continuous and binary responses in quantile regression. The methods are illustrated using several simulation studies and a microarray study. The simulation studies and the microarray study indicate that the proposed approach performs well.     

Efficient Robust Estimation for Linear Models with Missing Response at Random

Coefficient estimation in linear regression models with missing data is routinely carried out in the mean regression framework. However, the mean regression theory breaks down if the error variance is infinite. In addition, correct specification of the likelihood function for existing imputation approach is often challenging in practice, especially for skewed data. In this paper, we develop a novel composite quantile regression and a weighted quantile average estimation procedure for parameter estimation in linear regression models when some responses are missing at random. Instead of imputing the missing response by randomly drawing from its conditional distribution, we propose to impute both missing and observed responses by their estimated conditional quantiles given the observed data and to use the parametrically estimated propensity scores to weigh check functions that define a regression parameter. Both estimation procedures are resistant to heavy‐tailed errors or outliers in the response and can achieve nice robustness and efficiency. Moreover, we propose adaptive penalization methods to simultaneously select significant variables and estimate unknown parameters. Asymptotic properties of the proposed estimators are carefully investigated. An efficient algorithm is developed for fast implementation of the proposed methodologies. We also discuss a model selection criterion, which is based on an IC_Q ‐type statistic, to select the penalty parameters. The performance of the proposed methods is illustrated via simulated and real data sets.     

Gibbs sampling methods for Bayesian quantile regression

This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.     

Doubly robust weighted composite quantile regression based on SCAD-L2

In this article, a robust variable selection procedure based on the weighted composite quantile regression (WCQR) is proposed. Compared with the composite quantile regression (CQR), WCQR is robust to heavy-tailed errors and outliers in the explanatory variables. For the choice of the weights in the WCQR, we employ a weighting scheme based on the principal component method. To select variables with grouping effect, we consider WCQR with SCAD-L₂ penalization. Furthermore, under some suitable assumptions, the theoretical properties, including the consistency and oracle property of the estimator, are established with a diverging number of parameters. In addition, we study the numerical performance of the proposed method in the case of ultrahigh-dimensional data. Simulation studies and real examples are provided to demonstrate the superiority of our method over the CQR method when there are outliers in the explanatory variables and/or the random error is from a heavy-tailed distribution.     

Model selection in quantile regression models

Lasso methods are regularisation and shrinkage methods widely used for subset selection and estimation in regression problems. From a Bayesian perspective, the Lasso-type estimate can be viewed as a Bayesian posterior mode when specifying independent Laplace prior distributions for the coefficients of independent variables [32 T. Park, G. Casella, The Bayesian Lasso, J. Amer. Statist. Assoc. 103 (2008), pp. 681–686. doi: 10.1198/016214508000000337[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. A scale mixture of normal priors can also provide an adaptive regularisation method and represents an alternative model to the Bayesian Lasso-type model. In this paper, we assign a normal prior with mean zero and unknown variance for each quantile coefficient of independent variable. Then, a simple Markov Chain Monte Carlo-based computation technique is developed for quantile regression (QReg) models, including continuous, binary and left-censored outcomes. Based on the proposed prior, we propose a criterion for model selection in QReg models. The proposed criterion can be applied to classical least-squares, classical QReg, classical Tobit QReg and many others. For example, the proposed criterion can be applied to rq(), lm() and crq() which is available in an R package called Brq. Through simulation studies and analysis of a prostate cancer data set, we assess the performance of the proposed methods. The simulation studies and the prostate cancer data set analysis confirm that our methods perform well, compared with other approaches.     

Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

Small area estimation of the mean using non-parametric M-quantile regression: a comparison when a linear mixed model does not hold

The demand for reliable statistics in subpopulations, when only reduced sample sizes are available, has promoted the development of small area estimation methods. In particular, an approach that is now widely used is based on the seminal work by Battese et al. [An error-components model for prediction of county crop areas using survey and satellite data, J. Am. Statist. Assoc. 83 (1988), pp. 28–36] that uses linear mixed models (MM). We investigate alternatives when a linear MM does not hold because, on one side, linearity may not be assumed and/or, on the other, normality of the random effects may not be assumed. In particular, Opsomer et al. [Nonparametric small area estimation using penalized spline regression, J. R. Statist. Soc. Ser. B 70 (2008), pp. 265–283] propose an estimator that extends the linear MM approach to the case in which a linear relationship may not be assumed using penalized splines regression. From a very different perspective, Chambers and Tzavidis [M-quantile models for small area estimation, Biometrika 93 (2006), pp. 255–268] have recently proposed an approach for small-area estimation that is based on M-quantile (MQ) regression. This allows for models robust to outliers and to distributional assumptions on the errors and the area effects. However, when the functional form of the relationship between the qth MQ and the covariates is not linear, it can lead to biased estimates of the small area parameters. Pratesi et al. [Semiparametric M-quantile regression for estimating the proportion of acidic lakes in 8-digit HUCs of the Northeastern US, Environmetrics 19(7) (2008), pp. 687–701] apply an extended version of this approach for the estimation of the small area distribution function using a non-parametric specification of the conditional MQ of the response variable given the covariates [M. Pratesi, M.G. Ranalli, and N. Salvati, Nonparametric m-quantile regression using penalized splines, J. Nonparametric Stat. 21 (2009), pp. 287–304]. We will derive the small area estimator of the mean under this model, together with its mean-squared error estimator and compare its performance to the other estimators via simulations on both real and simulated data.     

General composite quantile regression: Theory and methods

Abstract

In this article, we propose a new regression method called general composite quantile regression (GCQR) which releases the unrealistic finite error variance assumption being imposed by the traditional least squares (LS) method. Unlike the recently proposed composite quantile regression (CQR) method, our proposed GCQR allows any continuous non-uniform density/weight function. As a result, determination of the number of uniform quantile positions is not required. Most importantly, the proposed GCQR criterion can be readily transformed to a linear programing problem, which substantially reduces the computing time. Our theoretical and empirical results show that the GCQR is generally efficient than the CQR and LS if the weight function is appropriately chosen. The oracle properties of the penalized GCQR are also provided. Our simulation results are consistent with the derived theoretical findings. A real data example is analyzed to demonstrate our methodologies.     

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 Lasso-mixed quantile regression

In this paper, we discuss the regularization in linear-mixed quantile regression. A hierarchical Bayesian model is used to shrink the fixed and random effects towards the common population values by introducing an l₁ penalty in the mixed quantile regression check function. A Gibbs sampler is developed to simulate the parameters from the posterior distributions. Through simulation studies and analysis of an age-related macular degeneration (ARMD) data, we assess the performance of the proposed method. The simulation studies and the ARMD data analysis indicate that the proposed method performs well in comparison with the other approaches.     

Composite kernel quantile regression

The composite quantile regression (CQR) has been developed for the robust and efficient estimation of regression coefficients in a liner regression model. By employing the idea of the CQR, we propose a new regression method, called composite kernel quantile regression (CKQR), which uses the sum of multiple check functions as a loss in reproducing kernel Hilbert spaces for the robust estimation of a nonlinear regression function. The numerical results demonstrate the usefulness of the proposed CKQR in estimating both conditional nonlinear mean and quantile functions.     

Adaptive composite quantile regressions and their asymptotic relative efficiency

The composite quantile regression (CQR for short) provides an efficient and robust estimation for regression coefficients. In this paper we introduce two adaptive CQR methods. We make two contributions to the quantile regression literature. The first is that, both adaptive estimators treat the quantile levels as realizations of a random variable. This is quite different from the classic CQR in which the quantile levels are typically equally spaced, or generally, are treated as fixed values. Because the asymptotic variances of the adaptive estimators depend upon the generic distribution of the quantile levels, it has the potential to enhance estimation efficiency of the classic CQR. We compare the asymptotic variance of the estimator obtained by the CQR with that obtained by quantile regressions at each single quantile level. The second contribution is that, in terms of relative efficiency, the two adaptive estimators can be asymptotically equivalent to the CQR method as long as we choose the generic distribution of the quantile levels properly. This observation is useful in that it allows to perform parallel distributed computing when the computational complexity issue arises for the CQR method. We compare the relative efficiency of the adaptive methods with respect to some existing approaches through comprehensive simulations and an application to a real-world problem.     

An extended fatigue life distribution

A five-parameter extended fatigue life model called the McDonald–Birnbaum–Saunders (McBS) distribution is proposed. It extends the Birnbaum–Saunders and beta Birnbaum–Saunders [G.M. Cordeiro and A.J. Lemonte, The β-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Comput. Statist. Data Anal. 55 (2011), pp. 1445–1461] distributions and also the new Kumaraswamy–Birnbaum–Saunders distribution. We obtain the ordinary moments, generating function, mean deviations and quantile function. The method of maximum likelihood is used to estimate the model parameters and its potentiality is illustrated with an application to a real fatigue data set. Further, we propose a new extended regression model based on the logarithm of the McBS distribution. This model can be very useful to the analysis of real data and could give more realistic fits than other special regression models.