138 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

New Bayesian  Lasso Composite Quantile Regression 

Fadel Hamid Hadi Alhusseini* 

Department of Statistics and Economic Informatics, University of Craiova, Romania University of Al-Qadiseya , 

Iraq 

Email: fadhelfadhel222@yahoo.com 

 

 

 

Abstract 

In this paper, we propose a new Bayesian lasso inference scheme for variable selection in composite quantile 

regression model (C Quantile Reg). The suggested approach is to construct a hierarchical structure within the 

Gibbs sampling under the assumption that the residual term comes from skew Laplace distribution (asymmetric 

Laplace distribution) and  assign scale mixture uniform (SMU) as prior distributions on the coefficients of 

composite quantile regression model. Our proposed method was compared to some other existing methods by 

testing the performance of these methods through simulation studies and real data examples. 

Keywords: New Bayesian Lasso; Posterior distributions; composite Quantile regression; Scale mixture of 
uniform. 

1. Introduction  

Since the seminal pioneering work of [11], quantile regression (Q Reg) has become more and more popular in 

numerous fields of science, for instance, the microarray study [25], agricultural economics [10], ecological 

studies [8], growth chart [26], etc. In addition, the quantile regression has good properties and it is a suitable 

regression model to non-normal errors, as quantile regression is more robust compared to ordinary least-square 

(OLS) regression [12]. QReg is more flexible, thus, it offers an extensive coverage of the response variable and 

its covariates correlations QReg operates without an assumption about the random error, providing greater 

statistical efficiency than traditional regression models when the error is non-normal. So far, QReg has proven 

to be an enhanced model of the conventional regression model.   

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

139 
 

The linear quantile regression model assumes that the response variable 𝒚𝒚𝒊𝒊 written as 

𝑦𝑦𝑖𝑖 = 𝛼𝛼𝜃𝜃 + 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽𝜃𝜃 + 𝑢𝑢𝑖𝑖                        𝑖𝑖 = 1,2, … ,𝑛𝑛                                           (1) 

where 𝑦𝑦𝑖𝑖   [𝑖𝑖 = 1,2, … . .𝑛𝑛] is response variable  

𝜃𝜃 is level quantile, and 𝜃𝜃  belongs to the open interval (0,1). 𝜃𝜃 ∈ (0,1) 

𝛼𝛼𝜃𝜃 is the intercept. 

𝛽𝛽𝜃𝜃 is vector of unknown parameters 

𝑥𝑥𝑖𝑖𝑇𝑇 is vector of covariates 

𝑢𝑢𝑖𝑖    is the random error term with 𝜃𝜃-th quantile equal to zero 

The parameters 𝛼𝛼𝜃𝜃 and 𝛽𝛽𝜃𝜃 to the quantile regression model are estimated by solving the following equation: 

 �𝜌𝜌𝜃𝜃

𝑛𝑛

𝑖𝑖=1
𝛼𝛼𝜃𝜃,𝛽𝛽𝜃𝜃
𝑚𝑚𝑖𝑖𝑛𝑛  (𝑦𝑦𝑖𝑖 − 𝛼𝛼𝜃𝜃 − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽𝜃𝜃)                                               (2) 

where 𝜌𝜌𝜃𝜃(𝑢𝑢) is the check (loss) function  

𝜌𝜌𝜃𝜃(𝑢𝑢) = �
𝜃𝜃𝑢𝑢                                       𝑖𝑖𝑖𝑖 𝑢𝑢 ≥ 0           

−(1 − 𝜃𝜃)𝑢𝑢                            𝑖𝑖𝑖𝑖 𝑢𝑢 < 0                       (3)     

Since equation (2) is not differentiable at the origin, there is no exact form of the solution for equation (2) [15]. 

The minimization of equation (2) can be achieved by a linear programming algorithm [13]. A Bayesian 

approach enables  the exact estimation of quantile regression parameters.  One important thing in building a 

regression model is the selection of the active covariates. The selection process aims to increase the prediction 

accuracy and to get high interpretation [4]. Recently, there has been considerable attention on regression models 

that include all covariates and use informative priors to shrink inactive regression coefficients toward zero 

exactly, for instance, the Lasso method [23], the adaptive Lasso, proposed by [26], dantzig selector [7], Lasso 

QReg [17] and adaptive Lasso QReg [28]. A comprehensive account of these and other recent methods can be 

found in the work of [24]. Similarly, from a Bayesian framework, [20] proposed Bayesian Lasso for traditional 

linear regression models by specifying scale mixture of normal (SMN) prior distributions. Reference [21] 

proposed Bayesian adaptive Lasso by using different shrinkage parameters. In later approaches, Reference [18] 

suggested Bayesian Lasso QReg and [4] proposed Bayesian adaptive Lasso Qreg. Some further extensions of 

the Lasso Qreg models have also been suggested by [5,1,27,29,2], among others. All aforementioned methods 

were focused on a single quantile level and scale mixture of normal (SMN) priors. [30,6].; proposed methods 

that focused on composite quantile regression model. [9] proposed a Bayesian approach of composite quantile 

regression by using scale mixture of normal (SMN) priors. In this paper our proposed method develops a 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

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Bayesian approach for regularization in linear composite QReg by assigning the scale mixture of uniform 

(SMU) formulation of the Laplace density. This approach was provided by [19]. In other word, in this paper, we 

propose a new formulation for Bayesian lasso composite QReg by using the scale mixture of uniform (SMU) 

formulation. The following sections of this paper are arranged in the following order: section 2 presents New 

Bayesian lasso composite QReg. In section 3, we perform simulation studies. In section 4, we show a real data 

example, and the conclusions and  recommendations  are included in section 5. 

2. New Bayesian Lasso Composite QReg 

2.1. Bayesian Composite QReg  

The quantile regression model is focused on modeling the relationship between the response variable and a set 

of covariates by single quantile level. From known, there are infinite quantile regression lines within 𝜃𝜃 ∈ (0,1) 

or 0 < 𝜃𝜃1,𝜃𝜃2, … . .𝜃𝜃𝐻𝐻 < 1 where H represents different quantiles. But choosing the informative quantile line is 

challenging and to overcome this problem, the information is used over all quantile lines. [30] introduced the 

mathematical formula to composite quantile regression model, as follows:  

𝛼𝛼𝜃𝜃1 ,𝛼𝛼𝜃𝜃2 , … … ,𝛼𝛼𝜃𝜃ℎ ,𝛽𝛽 =  ��𝜌𝜌𝜃𝜃ℎ(𝑦𝑦𝑖𝑖 − 𝛼𝛼ℎ

𝑛𝑛

𝑖𝑖=1

𝐻𝐻

ℎ=1
𝛼𝛼𝜃𝜃1 ,𝛼𝛼𝜃𝜃2 ,……,𝛼𝛼𝜃𝜃ℎ

𝑀𝑀𝑖𝑖𝑛𝑛            − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽)                     (4) 

where 

 ℎ = 1,2, … . ,𝐻𝐻 𝑎𝑎𝑛𝑛𝑎𝑎 𝑖𝑖 = 1,2, … … ,𝑛𝑛 

Hence, the check(loss) function of composite different quantiles takes the following form:  

𝜌𝜌𝜃𝜃ℎ(𝑢𝑢) = �
𝜃𝜃ℎ𝑢𝑢                                       𝑖𝑖𝑖𝑖 𝑢𝑢 ≥ 0           

−(1 − 𝜃𝜃ℎ)𝑢𝑢                            𝑖𝑖𝑖𝑖 𝑢𝑢 < 0                                                         (5)     

The equation (4) is also not differentiable at 0, and the minimization of (4) can be achieved by a linear 

programming algorithm [14]. It is possible to estimate the parameters of composite quantile regression by using 

a Bayesian approach through a new Gibbs sampler and this is our proposed method to posterior distributions. 

[28] noted that the random error of quantile is close to asymmetric Laplace distribution (ALD). 

 

𝑖𝑖𝐴𝐴𝐴𝐴𝐴𝐴(𝑢𝑢|𝜃𝜃ℎ) =
𝜃𝜃ℎ(1 − 𝜃𝜃ℎ) 

𝜎𝜎
 exp�𝜎𝜎−1𝜌𝜌𝜃𝜃ℎ(𝑢𝑢)�                              −∞ <  𝑥𝑥 < ∞                   (6) 

The function of asymmetric Laplace distribution with a scale parameter equal to 1 is 

  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

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𝑖𝑖𝐴𝐴𝐴𝐴𝐴𝐴(𝑢𝑢|𝜎𝜎,𝜃𝜃ℎ) = 𝜃𝜃ℎ(1 − 𝜃𝜃ℎ) exp�𝜌𝜌𝜃𝜃ℎ(𝑢𝑢)�                                  −∞ <  𝑥𝑥 < ∞                 (7) 

With mean 𝐸𝐸(𝑢𝑢) = (1−2𝜃𝜃ℎ  )
𝜃𝜃ℎ(1−𝜃𝜃ℎ)

  and the variance 𝑣𝑣𝑎𝑎𝑣𝑣 (𝑢𝑢) = �1−2𝜃𝜃ℎ  −2𝜃𝜃ℎ
2�

𝜃𝜃ℎ(1−𝜃𝜃ℎ)
 

The joint distribution of response variable = (𝑦𝑦1,𝑦𝑦2, … … ,𝑦𝑦𝑛𝑛)𝑇𝑇  , given  𝑋𝑋 = (𝑥𝑥1, 𝑥𝑥2, … . , 𝑥𝑥𝑛𝑛)𝑇𝑇 , 𝛼𝛼 =

�𝛼𝛼𝜃𝜃1 ,𝛼𝛼𝜃𝜃2 , … … ,𝛼𝛼𝜃𝜃ℎ�
𝑇𝑇
 and  𝛽𝛽 = (𝛽𝛽1,𝛽𝛽2, … . .𝛽𝛽𝑘𝑘)𝑇𝑇 for composite quantile regression is: 

𝑖𝑖𝐴𝐴𝐴𝐴𝐴𝐴(𝑦𝑦|𝑋𝑋,𝛼𝛼,𝛽𝛽) = �𝜃𝜃ℎ𝑛𝑛(1 − 𝜃𝜃ℎ𝑛𝑛) exp��𝜌𝜌𝜃𝜃ℎ(𝑦𝑦 − 𝛼𝛼ℎ − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽)
𝑛𝑛

𝑖𝑖=1

�
𝐻𝐻

ℎ=1

                           (8)       

Hence, maximizing the likelihood function of response variable 𝑦𝑦 in equation (8) is equivalent to minimizing 

equation (4). It is very difficult to use the likelihood function of response variable 𝑦𝑦 directly, because of the 

mixture of H components. Following [9], we employ a cluster assignment matrix 𝐶𝐶 with its (𝑖𝑖, ℎ)𝑡𝑡ℎ element, 𝐶𝐶𝑖𝑖ℎ 

taking two values: one (𝐶𝐶𝑖𝑖ℎ = 1) if 𝑖𝑖𝑖𝑖ℎ subject belongs to the ℎ 𝑖𝑖ℎ cluster, or zero (𝐶𝐶𝑖𝑖ℎ = 0) if the element 𝐶𝐶𝑖𝑖ℎ is 

treated like missing value. Therefore, our likelihood takes the following form:  

𝑖𝑖𝐴𝐴𝐴𝐴𝐴𝐴(𝑦𝑦|𝑋𝑋,𝛼𝛼,𝛽𝛽) = ���𝜃𝜃ℎ(1 − 𝜃𝜃ℎ) exp�𝜌𝜌𝜃𝜃ℎ(𝑦𝑦 − 𝛼𝛼ℎ − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽)��
𝐶𝐶𝑖𝑖ℎ

𝑛𝑛

𝑖𝑖=1

𝐻𝐻

ℎ=1

                    (9)       

Recently, [16] reformulated the distribution of the (ALD) as a mixture of normal distributions (SMN). More 

specifically, the random error is equal to   

𝑢𝑢 = 𝜁𝜁1ℎ𝑧𝑧𝑖𝑖 + 𝜁𝜁2ℎ�𝑧𝑧𝑖𝑖  𝜖𝜖, where 𝑧𝑧𝑖𝑖~𝐸𝐸𝑥𝑥𝐸𝐸[𝜃𝜃ℎ(1 − 𝜃𝜃ℎ)] and 𝜖𝜖  is distributed standard normal distribution, 𝜖𝜖~𝑁𝑁(0,1). 

Therefore, the random error 𝑢𝑢 is distributed normal distribution with mean ((1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖  and variance (2𝑧𝑧𝑖𝑖), 

𝑢𝑢~𝑁𝑁(�(1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖 , 2𝑧𝑧𝑖𝑖� . As a result, the response variable 𝑦𝑦  is distributed normal distribution with mean 

(𝛼𝛼ℎ + 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽 + (1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖) and variance (2𝑧𝑧𝑖𝑖). 

,𝑦𝑦𝑖𝑖|𝛼𝛼ℎ,𝛽𝛽, 𝑧𝑧𝑖𝑖~𝑁𝑁(𝛼𝛼ℎ + 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽 + (1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖 , 2𝑧𝑧𝑖𝑖). According to the formulation of [16], our Bayesian composite 

quantile regression can be written as: 

𝑖𝑖(𝑦𝑦|𝑋𝑋, 𝑧𝑧𝑖𝑖 ,𝛽𝛽,𝛼𝛼ℎ ,𝐶𝐶) = ��
1

�4𝜋𝜋𝑧𝑧𝑖𝑖
�
𝐶𝐶𝑖𝑖ℎ𝑛𝑛

𝑖𝑖=1

𝑒𝑒𝑥𝑥𝐸𝐸 �−
1
2
��

𝐶𝐶𝑖𝑖ℎ(𝑦𝑦 − 𝛼𝛼ℎ − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽 − (1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖)
2𝑧𝑧𝑖𝑖

𝐻𝐻

ℎ=1

𝑛𝑛

𝑖𝑖=1

�                  (10) 

Under the of formulation equation (10), the composite quantile regression coefficients have good features for 

constructing a good and efficient Gibbs sampler algorithm. 

2.2. New Bayesian Lasso Composite QReg (new B C QReg) 

[23] proposed a new method of estimation and variables selection in linear regression model,  known as lasso 

(Least Absolute Shrinkage and Selection Operator) . In the same context  [23] mentioned Bayesian lasso in 

linear regression model is possible under assigned Laplace distribution as priors for the 𝛽𝛽𝑗𝑗. [12] proposed lasso 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

142 
 

quantile regression that takes the following formula: 

 �𝜌𝜌𝜃𝜃

𝑛𝑛

𝑖𝑖=1
,𝛽𝛽𝜃𝜃    
𝑚𝑚𝑖𝑖𝑛𝑛  (𝑦𝑦𝑖𝑖 − 𝛼𝛼𝜃𝜃 − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽) + 𝜆𝜆�|𝛽𝛽|�                                       (11) 

where 𝜆𝜆(𝜆𝜆 ≥ 0) is the tuning parameter.  

[18] suggested Bayesian Lasso for linear quantile regression model by assigning Laplace prior for the 𝛽𝛽𝑗𝑗.The 

probability density function takes the specification of the form: 

𝐸𝐸�𝛽𝛽𝑗𝑗�𝜆𝜆� =
𝜆𝜆
2

 𝑒𝑒𝑥𝑥𝐸𝐸[−𝜆𝜆|𝛽𝛽|]                                                                       (12) 

It is difficult to use equation (12) directly in the estimation of the model parameters. Therefore, most researchers 

in the field of Bayesian lasso regression models use the transformation Laplace distribution to the scale mixture 

normal. The works of [20]; [17]; [4] and [9] are relevant. In this paper, we develop an alternative hierarchical 

Bayesian model by using a new lasso technique. According to the proposal of [19] the Laplace prior distribution 

on coefficients (βj) can be written as: 

𝜆𝜆
2

 𝑒𝑒𝑥𝑥𝐸𝐸[−𝜆𝜆|𝛽𝛽|] = �
1

2𝑠𝑠𝑗𝑗
 
𝜆𝜆2

Γ(2) 𝑠𝑠𝑗𝑗2−1 𝑒𝑒𝑥𝑥𝐸𝐸�−𝜆𝜆𝑠𝑠𝑗𝑗� 𝑎𝑎𝑠𝑠𝑗𝑗                                              (13)  
.

𝑠𝑠𝑗𝑗>|𝛽𝛽𝑗𝑗|

    

= �  
𝜆𝜆2

2
  𝑒𝑒𝑥𝑥𝐸𝐸�−𝜆𝜆𝑠𝑠𝑗𝑗� 𝑎𝑎𝑠𝑠𝑗𝑗                                              

.

𝑠𝑠𝑗𝑗>|𝛽𝛽𝑗𝑗|

 

where  Γ(2) = (2 − 1)! = 1 

 

= 𝜆𝜆
2

2
   �

𝑒𝑒𝑒𝑒𝑒𝑒�−𝜆𝜆𝑠𝑠𝑗𝑗�

−𝜆𝜆
�

|𝛽𝛽𝑗𝑗|

∞
=  𝜆𝜆

2

2
 �𝑒𝑒𝑒𝑒𝑒𝑒{−𝜆𝜆∞}

−𝜆𝜆
+ 𝑒𝑒𝑒𝑒𝑒𝑒{−𝜆𝜆|𝛽𝛽|}

−𝜆𝜆
� = 𝜆𝜆2

2
 �0 + 𝑒𝑒𝑒𝑒𝑒𝑒{−𝜆𝜆|𝛽𝛽|}

−𝜆𝜆
� = 𝜆𝜆

2
 𝑒𝑒𝑥𝑥𝐸𝐸[−𝜆𝜆|𝛽𝛽|] 

where sj is a mixing variable. Our Bayesian hierarchical model can be formulated as below after assigning 

Gamma priors on 𝜆𝜆.  

𝑖𝑖(𝑦𝑦|𝑋𝑋, 𝑧𝑧𝑖𝑖 ,𝛽𝛽,𝛼𝛼ℎ,𝐶𝐶) = ��
1

�4𝜋𝜋𝑧𝑧𝑖𝑖
�
𝐶𝐶𝑖𝑖ℎ𝑛𝑛

𝑖𝑖=1

𝑒𝑒𝑥𝑥𝐸𝐸 �−
1
2
��

𝐶𝐶𝑖𝑖ℎ(𝑦𝑦 − 𝛼𝛼ℎ − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽 − (1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖)
2𝑧𝑧𝑖𝑖

𝐻𝐻

ℎ=1

𝑛𝑛

𝑖𝑖=1

� 

         𝐸𝐸(𝛼𝛼ℎ) ∝ 1 

        𝑧𝑧𝑖𝑖~𝑒𝑒𝑥𝑥𝐸𝐸{𝜃𝜃ℎ(1 − 𝜃𝜃ℎ)}, 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

143 
 

        𝛽𝛽𝑗𝑗|𝑠𝑠𝑗𝑗~𝑈𝑈𝑛𝑛𝑖𝑖𝑖𝑖𝑈𝑈𝑣𝑣𝑈𝑈 �−𝑠𝑠𝑗𝑗 , 𝑠𝑠𝑗𝑗�, 

        𝑠𝑠𝑗𝑗|𝜆𝜆 ~𝐺𝐺𝑎𝑎𝑈𝑈𝑈𝑈𝑎𝑎(2, 𝜆𝜆) ,                                                                                               (14) 

        𝜆𝜆~𝜆𝜆𝑎𝑎−1𝑒𝑒𝑥𝑥𝐸𝐸(−𝑏𝑏𝜆𝜆) . 

where 𝑎𝑎, 𝑏𝑏 are hyperparameter. 

2.2.1. Posterior Computation Inference 

The Bayesian theory is used for the conditional posterior distribution for the parameters of the model by 

combining two different sources of information.  

The first source is the likelihood function of response variable 𝑦𝑦 in equation (10) and the second source is a set 

of prior distributions to model parameters in equation (14).  

The Bayesian hierarchical in (10) and (14) produces posterior distributions �𝛼𝛼ℎ,𝛽𝛽, 𝑧𝑧 = (𝑧𝑧1, 𝑧𝑧2, … . , 𝑧𝑧𝑛𝑛)𝑇𝑇 , 𝑠𝑠 =

(𝑠𝑠1, 𝑠𝑠2, … . , 𝑠𝑠𝑒𝑒)𝑇𝑇 , 𝜆𝜆  𝑎𝑎𝑛𝑛𝑎𝑎 𝐶𝐶�. A simple and efficient Gibbs sampler algorithm is as follows: 

- Updating 𝑧𝑧𝑖𝑖  -the full conditional distribution of each 𝑧𝑧𝑖𝑖   foe 𝑖𝑖 = 1,2, … … 𝑛𝑛,  𝐼𝐼𝑛𝑛𝑣𝑣𝐺𝐺(𝛿𝛿𝑖𝑖𝑇𝑇 ,𝜑𝜑𝑖𝑖𝑇𝑇)  , where 𝜑𝜑𝑇𝑇 =

∑ 𝑐𝑐𝑖𝑖ℎ
𝐻𝐻
ℎ=1   
2

 and  

𝛿𝛿𝑖𝑖𝑇𝑇 = �
∑ 𝑐𝑐𝑖𝑖ℎ𝐻𝐻
ℎ=1

2
∑ 𝑐𝑐𝑖𝑖ℎ (𝑦𝑦𝑖𝑖 − 𝛼𝛼ℎ − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽)2𝐻𝐻
ℎ=1

2

= �
∑ 𝑐𝑐𝑖𝑖ℎ𝐻𝐻
ℎ=1

∑ 𝑐𝑐𝑖𝑖ℎ (𝑦𝑦𝑖𝑖 − 𝛼𝛼ℎ − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽)2𝐻𝐻
ℎ=1

�  

- Updating 𝛼𝛼ℎ - the full conditional distribution of each 𝛼𝛼ℎ  foe ℎ = 1,2, … …𝐻𝐻,   𝑖𝑖𝑠𝑠 𝑁𝑁(𝛼𝛼�ℎ,𝜎𝜎�ℎ2), 

𝛼𝛼�ℎ = �𝜎𝜎�ℎ
2 ∑ 𝑐𝑐𝑖𝑖ℎ [𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽 − (1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖)    𝑛𝑛

𝑖𝑖=1
2𝑧𝑧𝑖𝑖
� �,  𝜎𝜎�ℎ2 = �∑ 𝑐𝑐𝑖𝑖ℎ

𝑛𝑛
𝑖𝑖=1
2𝑧𝑧𝑖𝑖

�
−1

 

- Updating 𝛽𝛽𝑗𝑗 - the full conditional distribution of each 𝛽𝛽𝑗𝑗   for 𝑗𝑗 = 1,2, … … 𝑘𝑘,   𝑖𝑖𝑠𝑠 𝑁𝑁(𝛽𝛽�𝑗𝑗 ,𝜎𝜎�𝑗𝑗2), where: 

𝛽𝛽�𝑗𝑗 = �
𝜎𝜎�𝑗𝑗2 ∑ ∑ 𝑐𝑐𝑖𝑖ℎ𝑥𝑥𝑖𝑖𝑗𝑗�𝑇𝑇𝑖𝑖 − 𝛼𝛼ℎ − (1 − 2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖 − ∑ 𝑥𝑥𝑖𝑖𝑗𝑗𝑙𝑙≠𝑗𝑗 𝛽𝛽𝑗𝑗�𝐻𝐻

ℎ=1
𝑛𝑛
𝑖𝑖=1

2𝑧𝑧𝑖𝑖
� �  𝐼𝐼�|𝛽𝛽𝑗𝑗| ≤ 𝑠𝑠𝑗𝑗� 

and 𝜎𝜎�𝑗𝑗2 = �
∑ ∑ 𝑐𝑐𝑖𝑖ℎ𝑒𝑒𝑖𝑖𝑗𝑗

2𝐻𝐻
ℎ=1

𝑛𝑛
𝑖𝑖=1

2𝑧𝑧𝑖𝑖
+ 𝑠𝑠𝑗𝑗−1�

−1

. 

Updating 𝑠𝑠𝑗𝑗 -the full conditional distribution of 𝑠𝑠𝑗𝑗 is a left-truncated exponential distribution given by 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

144 
 

𝑠𝑠𝑗𝑗~𝐸𝐸𝑥𝑥𝐸𝐸(𝜆𝜆)𝐼𝐼�𝑠𝑠𝑗𝑗 > |𝛽𝛽𝑗𝑗|� . for 𝑗𝑗 = 1,2, … … 𝑘𝑘,. 

 Updating 𝑠𝑠𝑗𝑗 can be done by using inversion method [19] as follows: 

1-Update 𝑠𝑠𝑗𝑗∗ from 𝐸𝐸𝑥𝑥𝐸𝐸(𝜆𝜆) 

2- Set 𝑠𝑠𝑗𝑗 = 𝑠𝑠𝑗𝑗∗ + |𝛽𝛽𝑗𝑗|.     

 Updating 𝜆𝜆𝑗𝑗 - the full conditional distribution of each 𝜆𝜆𝑗𝑗  for 𝑗𝑗 = 1,2, … … 𝑘𝑘,.is Gamma �𝑎𝑎 + 2𝑘𝑘, 𝑏𝑏 + ∑ |𝛽𝛽𝑗𝑗|𝑘𝑘
𝑗𝑗=1 �.  

Updating 𝐶𝐶 - the full conditional distribution of each 𝐶𝐶𝑖𝑖 = (𝐶𝐶𝑖𝑖1,𝐶𝐶𝑖𝑖2,       ,𝐶𝐶𝑖𝑖ℎ)𝑇𝑇   is multinomial distribution with 

�̂�𝐸ℎ =

exp 

⎝

⎜
⎛−�𝑦𝑦𝑖𝑖−𝛼𝛼ℎ−𝑒𝑒𝑖𝑖

𝑇𝑇𝛽𝛽−(1−2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖�
2

4𝑧𝑧𝑖𝑖
�

⎠

⎟
⎞

∑ exp 

⎝

⎜
⎛−�𝑦𝑦𝑖𝑖−𝛼𝛼ℎ−𝑒𝑒𝑖𝑖

𝑇𝑇𝛽𝛽−(1−2𝜃𝜃ℎ)𝑧𝑧𝑖𝑖�
2

4𝑧𝑧𝑖𝑖
�

⎠

⎟
⎞𝑀𝑀

𝑚𝑚=1

 

After updating the model parameters as illustrated above, an efficient Gibbs sampler results for building 

(MCMC) algorithm for parameters estimation [𝛼𝛼ℎ,𝛽𝛽, 𝑧𝑧, 𝑠𝑠, 𝜆𝜆  𝑎𝑎𝑛𝑛𝑎𝑎 𝐶𝐶]. Our algorithm is run for 11000 iterations. 

The first 1000 iterations were discarded as burn-in. 

3. Simulation studies 

In this section, the behavior of our proposed method is assessed by simulation studies. To evaluate our method, 

a comparison is performed, using a set of methods: Bayesian Lasso quantile regression proposed by [18] and 

classical frequentist approach using the R function rq() in the R package quantreg.  

The methods are assessed based on two criteria - median of mean absolute deviations (MMAD), where MMAD =

median mean ��𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽𝐸𝐸𝑠𝑠𝑡𝑡𝑖𝑖𝑚𝑚𝑎𝑎𝑡𝑡𝑒𝑒𝐸𝐸 − 𝑥𝑥𝑖𝑖𝑇𝑇𝛽𝛽𝑡𝑡𝑡𝑡𝑡𝑡𝑒𝑒��, and standard deviation (SD) of the MADs. In this approach, we 

chose a set 𝐻𝐻 = 3 (three quantile levels, as follows: 𝜃𝜃1 = 0.25 is the low quantile level, 𝜃𝜃2 = 0.55 is the middle 

quantile level and 𝜃𝜃3 = 0.85 is the high quantile level) for each simulation study.  

The error terms generated from four distributions are standard normal distribution 𝑁𝑁(0,1), 𝜒𝜒(3)
2 distribution with 

three degrees of freedom mixture normal distribution 𝑢𝑢𝑖𝑖~
1
2
𝑁𝑁 (0,1) + 1

2
𝑁𝑁(1,1) and mixture Laplace distribution 

𝑢𝑢𝑖𝑖~
1
2
𝐿𝐿𝑎𝑎𝐸𝐸𝐿𝐿𝑎𝑎𝑐𝑐𝑒𝑒 (0,1) + 1

2
𝐿𝐿𝑎𝑎𝐸𝐸𝐿𝐿𝑎𝑎𝑐𝑐𝑒𝑒(1,1).  

Our algorithm is run 11000 iterations and the first 1000 are discarded as the burn-in. To assess the performance 

of our proposed method, we used three approaches. 

3.1 The First Approach 



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145 
 

In this section, the simulated study is done with a very sparse case. This is clear from true parameters values, 

𝛽𝛽 = (1,0,0,0,0,0,0,0)𝑇𝑇. Therefore, the true model becomes as follows: 

𝑦𝑦𝑖𝑖 = 0 + 𝑥𝑥1𝑖𝑖 + 𝑢𝑢𝑖𝑖 ,        𝑖𝑖 = 1,2, … … . .100   

We simulate eight covariates ( 𝑥𝑥1𝑖𝑖 , 𝑥𝑥2𝑖𝑖 , 𝑥𝑥3𝑖𝑖 , 𝑥𝑥4𝑖𝑖 , 𝑥𝑥5𝑖𝑖 ,𝑥𝑥6𝑖𝑖 , 𝑥𝑥7𝑖𝑖 , 𝑥𝑥8𝑖𝑖)  from a multivariate normal distribution 

𝑋𝑋~𝑁𝑁8(𝜇𝜇, Σ), where 𝜇𝜇 is mean vector 𝜇𝜇 𝜖𝜖 𝑅𝑅𝑛𝑛 and Σ is covariance matrix with (Σ𝑒𝑒)𝑖𝑖𝑗𝑗 = (2−1)|𝑖𝑖−𝑗𝑗| . 

3.2 The Second Approach 

The simulated study is done with a very dense case. This is obvious from true parameters values 𝛽𝛽 =

(0.85, 0.85, 0.85, 0.85, 0.85, 0.85, 0.85, 0.85)𝑇𝑇.  

Therefore, the true model becomes as follows: 

𝑦𝑦𝑖𝑖 = 0.85𝑥𝑥1𝑖𝑖 + 0.85𝑥𝑥1𝑖𝑖 + 0.85𝑥𝑥2𝑖𝑖 + 0.85𝑥𝑥3𝑖𝑖 + 0.85𝑥𝑥4𝑖𝑖 + 0.85𝑥𝑥5𝑖𝑖 + 0.85𝑥𝑥6𝑖𝑖 + 0.85𝑥𝑥7𝑖𝑖 + 0.85𝑥𝑥7𝑖𝑖 + 𝑢𝑢𝑖𝑖 ,  

       𝑖𝑖 = 1,2, … … . .100   

We simulated eight covariates (𝑥𝑥1𝑖𝑖 ,  𝑥𝑥2𝑖𝑖 , 𝑥𝑥3𝑖𝑖 ,  𝑥𝑥4𝑖𝑖 ,  𝑥𝑥5𝑖𝑖 , 𝑥𝑥6𝑖𝑖 ,  𝑥𝑥7𝑖𝑖 ,  𝑥𝑥8𝑖𝑖)  from a multivariate normal distribution 

𝑋𝑋~𝑁𝑁8(𝜇𝜇, Σ), where 𝜇𝜇 is mean vector 𝜇𝜇 𝜖𝜖 𝑅𝑅𝑛𝑛 and Σ is covariance matrix with (Σ𝑒𝑒)𝑖𝑖𝑗𝑗 = (2−1)|𝑖𝑖−𝑗𝑗| . 

3.3 The Third Approach 

The simulated study is done with group structures. This is obvious from true parameters values 𝛽𝛽 =

[(0,0,0), (2,2,2), (0,0,0), (2,2,2), (0,0,0)]𝑇𝑇. Therefore, the true model takes the formula below: 

𝑦𝑦𝑖𝑖 = 2𝑥𝑥4𝑖𝑖 + 2𝑥𝑥5𝑖𝑖 + 2𝑥𝑥6𝑖𝑖 + 2𝑥𝑥10𝑖𝑖 + 2𝑥𝑥11𝑖𝑖 + 2𝑥𝑥12𝑖𝑖 + 𝑢𝑢𝑖𝑖 ,          𝑖𝑖 = 1,2, … … . .100 

where the covariates are simulated. 

𝑥𝑥1𝑖𝑖 , 𝑥𝑥2𝑖𝑖 ,𝑥𝑥3𝑖𝑖 , 𝑥𝑥4𝑖𝑖 , 𝑥𝑥5𝑖𝑖 , 𝑥𝑥6𝑖𝑖 , 𝑥𝑥7𝑖𝑖 , 𝑥𝑥8𝑖𝑖,𝑥𝑥9𝑖𝑖,𝑥𝑥10𝑖𝑖,𝑥𝑥11𝑖𝑖,𝑥𝑥12𝑖𝑖,𝑥𝑥13𝑖𝑖,𝑥𝑥14𝑖𝑖,𝑥𝑥15𝑖𝑖) according to a multivariate Gaussian distribution 

𝑋𝑋~𝑁𝑁15(𝜇𝜇, Σ), where 𝜇𝜇 is mean vector 𝜇𝜇 𝜖𝜖 𝑅𝑅𝑛𝑛 and Σ is covariance matrix with (Σ𝑒𝑒)𝑖𝑖𝑗𝑗 = (2−1)|𝑖𝑖−𝑗𝑗| . 

Table 1 shows the results of (MMAD) and (SD) averaged over 100 independent simulations for the three 

methods under comparison, via three quantile levels. The method proposed (the New Lasso CQ Reg) displays 

better results than the rq and 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁 methods, as can be seen in Table 1.  

In case of MMAD, for all considered distributions, the values are much smaller when New Lasso CQ Reg in 

employed, compared to rq and 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁 methods.  

In case of SD, the values are also smaller than for the other two methods, for different distributions. Taking into 

consideration these aspects, the New Lasso CQ Reg has superior performance compared to the other two 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

146 
 

methods under test.   

Table 1: Summary of MMAD and 𝑆𝑆𝑆𝑆 values. The results are averaged over one hundred simulated data set for 

the methods under comparison. 

  Error distributions  

 Methods  𝑁𝑁(0,1) 𝜒𝜒(3)
2  Normal mixture Laplace mixture 

First approach      

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 1.929 (0.870) 2.776(0.345) 2.513 (0.663) 2.067(1.013) 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 2.160(0.588) 3.154(0.264) 2.614 (0.566) 2.218(0.891) 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 2.513(0.661) 3.744(0.428) 2.959 (0.510) 2.678(0.772) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 1.411(0.599) 2.113(0.343) 1.847 (0.483) 1.476(0.821) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 1.932(0.606) 2.933(0.375) 2.434 (0.474) 2.031(0.781) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 2.415 (0.626) 3.653(0.414) 2.889 (0.442) 2.483 (0.758) 

 New Lasso CQ Reg 0.902(0.200) 0.560(0.174) 0.715 (0.215) 0.741(0.310) 

Second approach      

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 1.698 (0.171) 2.520(0.338) 2.018 (0.384) 2.007(0.156) 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 1.959 (0.258) 2.987 (0.409) 2.375 (0.295) 2.329 (0.161) 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 2.256 (0.268) 3.681 (0.948) 2.724 (0.432) 2.741 (0.301) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 1.264 (0.216) 1.858 (0.932) 1.546 (0.337) 1.431 (0.186) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 1.736 (0.258) 2.754 (0.287) 2.104 (0.318) 2.057 (0.235) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 2.142 (0.309) 3.560 (0.437) 2.592 (0.408) 2.647 (0.331) 

 New Lasso CQ Reg 0.765 (0.106) 0.703 (0.122) 0.621 (0.124) 0.596 (0.331) 

Third approach      

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 0.930 (0.841) 3.446 (0.410) 2.752(1.071) 2.813(0.689) 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 0.876 (0.667) 4.005 (0.425) 3.200 (1.020) 3.209(0.508) 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 0.837 (0.777) 4.845 (0.504) 3.927 (0.758) 3.777(0.631) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 0.851(0.769) 2.623 (0.357) 2.090 (1.020) 2.221(0.522) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 0.885(0.704) 3.594 (0.361) 2.923 (0.959) 2.981(0.551) 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 0.869(0.686) 4.707 (0.411) 3.851 (0.814) 3.719(0.683) 

 New Lasso CQ Reg 0.373(0.420) 1.089 (0.307) 1.496 (0.268) 1.157(0.352) 

 

The standard deviations of the MAD are mentioned in parentheses.  

The Table 2 provides additional criteria that represent the parameters estimation in direct way. As can be 

observed in the table, the parameters estimation through our proposed method (New Lasso CQ Reg) was very 

close to true parameters comparison with rq and 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁 methods.  

Therefore, the proposed method (New Lasso CQ Reg) has performed better than rq and 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁 methods.  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 29, No  1, pp 138-152 

147 
 

Table 2: Posterior means for simulation of the first approach via three quantile levels and four error distribution; 

the results are averaged over 100 simulations    

Error 

distribution 

Method 𝜷𝜷�𝟏𝟏 𝜷𝜷�𝟐𝟐 𝜷𝜷�𝟑𝟑 𝜷𝜷�𝟒𝟒 𝜷𝜷�𝟓𝟓 𝜷𝜷�𝟔𝟔 𝜷𝜷�𝟕𝟕 𝜷𝜷�𝟖𝟖 

𝑵𝑵(𝟎𝟎,𝟏𝟏) True parameters  1 0 0 0 0 0 0 0 

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 -0.882 1.807 -0.384 -0.157 0.878 0.192 0.237 0.340 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 -1.035 2.320 -0.717 -0.613 1.392 0.043 0.253 0.331 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 -2.135 3.572 -1.852 -0.402 1.707 0.130 0.377 0.114 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 -0.686 1.790 -0.481 0.105 0.978 0.238 0.475 0.366 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 -1.301 2.535 -1.061 -0.184 1.414 0.116 0.444 0.387 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 -2.230 3.333 -1.952 -0.186 1.642 0.374 0.126 0.464 

 New Lasso CQ Reg 0.970 0.017 0.002 0.121 0.009 0.003 0.044 0.145 

𝝌𝝌(𝟑𝟑)
𝟐𝟐  True parameters 1 0 0 0 0 0 0 0 

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 1.377 0.465 0.469 -0.784 1.954 1.366 -0.135 0.361 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 1.358 0.096 1.747 -2.080 1.971 1.936 -0.207 1.341 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 0.148 0.129 2.381 -1.942 3.008 2.265 -0.887 1.119 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 2.105 -0.021 0.992 -1.269 1.646 1.081 -0.307 0.662 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 1.383 0.009 1.663 -1.821 2.398 1.724 -0.637 0.987 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 0.198 0.322 2.366 -2.179 2.764 2.410 -1.041 1.075 

 New Lasso CQ Reg 1.242 0.065 0.285 0.342 0.183 0.283 0.218 0.327 

Mixture Normal True parameters 1 0 0 0 0 0 0 0 

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 1.275 0.288 0.290 -0.137 0.752 -0.234 1.014 -0.769 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 2.012 0.682 -0.410 0.283 1.378 -0.732 1.921 -1.356 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 1.539 1.335 -0.182 -0.472 1.582 -1.698 3.037 -1.573 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 2.489 0.631 -0.109 -0.083 0.856 -0.755 1.353 -0.901 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 2.116 0.885 -0.346 -0.036 1.271 -1.055 2.029 -1.186 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 1.715 1.286 -0.348 -0.234 1.579 -1.448 2.762 -1.413 

 New Lasso CQ Reg 1.244 0.121 0.026 0.238 0.131 0.160 0.234 0.070 

Mixture Laplace True parameters 1 0 0 0 0 0 0 0 

 𝑣𝑣𝑟𝑟𝜃𝜃1 = 0.25 5.050 0.120 -1.650 1.393 0.807 0.301 -1.553 0.523 

 𝑣𝑣𝑟𝑟𝜃𝜃2 = 0.55 5.046 -0.100 -1.558 2.040 0.929 -0.172 -1.443 0.450 

 𝑣𝑣𝑟𝑟𝜃𝜃3 = 0.85 5.554 -0.689 -0.967 2.502 0.951 -0.694 -1.950 0.669 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.25 5.265 -0.373 -0.609 1.167 -0.146 -0.971 -0.971 0.393 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.55 5.285 -0.584 -0.924 1.858 0.740 -0.288 -1.447 0.489 

 𝐿𝐿𝑎𝑎𝑠𝑠𝑠𝑠𝑈𝑈 𝑁𝑁𝜃𝜃1 = 0.85 5.346 -0.846 -0.999 2.426 0.943 -0.608 -1.941 0.618 

 New B Lasso CQ Reg 0.159 0.110 0.464 0.322 0.037 0.191 0.350 0.120 

 



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Figure 1 provides a clear vision about table 2. 

 

 

 

 

 

Figure 1: Bar chart summarizing the parameters of the studied methods close to true parameters 



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149 
 

4. Real data example  

In this section, we apply the three methods to prostate cancer data, first analysed by  [22]. It was also analyzed 

using the model selection in quantile regression model by [3].  

These data exist within "bayesQR" package in R. The sample size of prostate cancer data was of 97 

observations, the response variable is the level of prostate antigen referred to as (lpsa) and there are eight 

covariates.  

These covariates are 𝑥𝑥1  logarithm of cancer amount, referred to as (lcavol), 𝑥𝑥2  logarithm of the weight of 

prostate, referred to as (lweight), 𝑥𝑥3 age , 𝑥𝑥4 logarithm of the volume of benign enlargement of the prostate, 

referred to as (lbph), 𝑥𝑥5 seminal vesicle invasion, referred to as (svi), 𝑥𝑥6 logarithm of Capsular penetration in 

prostate cancer, referred to as (lcp ), 𝑥𝑥7 Gleason score, referred to as (gleason) and 𝑥𝑥8  percentage of Gleason 

scores 4 or 5, referred to as (pgg45).  

We applied a new Bayesian lasso composite quantile regression (New B Lasso CQ Reg) to the prostate cancer 

data and considered H=5 so that 𝜃𝜃ℎ = [0.16 , 0.33, 0.50  , 0.66  ,0.83]    𝑤𝑤ℎ𝑒𝑒𝑣𝑣𝑒𝑒 ℎ = 1,2,3,4,5)  𝑣𝑣𝑒𝑒𝑠𝑠𝐸𝐸𝑒𝑒𝑐𝑐𝑖𝑖𝑖𝑖𝑣𝑣𝑒𝑒𝐿𝐿𝑦𝑦 . 

Modeling the relationship between response variable and the covariates via the three methods is summarized in 

Table 3.  

Table 3:  Parameter estimates via the three methods in the comparison 

  rq      𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝑵𝑵   New B 

Lasso CQ 

Reg 

Variables 
     𝜃𝜃1
= 0.16 

    𝜃𝜃2
= 0.33 

   𝜃𝜃3
= 0.50 

   𝜃𝜃4
= 0.66 

   𝜃𝜃5
= 0.83 

   𝜃𝜃1 

= 0.16 

   𝜃𝜃2
= 0.33 

  𝜃𝜃3 

= 0.50 

   𝜃𝜃4
= 0.66 

   𝜃𝜃5
= 0.83 

M=5 

Intercept -0.521 -0.152 -0.057 0.170 0.519 -0.800 -0.305 -0.0003 0.291 0.763 0.459 

lcavol 0.742 0.607 0.543 0.513 0.633 0.648 0.607 0.571 0.563 0.559 0.241 

lweight 0.273 0.270 0.238 0.156 -0.007 0.227 0.245 0.219 0.185 0.162 0.155 

age -0.059 -0.169 -0.172 -0.133 -0.037 -0.093 -0.122 -0.140 -0.139 -0.119 0.152 

lbph 0.042 0.182 0.201 0.182 0.183 0.144 0.164 0.168 0.153 0.121 0.208 

svi 0.341 0.243 0.286 0.259 0.309 0.211 0.261 0.292 0.308 0.335 0.252 

lcp -0.349 -0.176 -0.158 -0.087 -0.112 -0.258 -0.223 -0.164 -0.101 -0.024 0.308 

gleason -0.041 0.046 0.127 0.032 -0.073 -0.023 0.005 0.032 0.040 0.041 0.114 

pgg45 0.318 0.153 0.099 0.115 0.085 0.214 0.195 0.147 0.106 0.071     0.108 

 

The parameter estimates in Table 3 are used to calculate the mean square error (MSE), as shown in table 4.  



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Table 4:  Mean square error (MSE) for the three methods 

Methods    𝜃𝜃1 = 0.16   𝜃𝜃2 = 0.33   𝜃𝜃3 = 0.50   𝜃𝜃4 = 0.66   𝜃𝜃5 = 0.83 

rq 9.487 7.401 6.912 5.820 5.358 

𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝑵𝑵 11.247 8.234 6.618 5.252 5.412 

New B Lasso CQ 

Reg 

5.077 

 

Table 4 shows the value of (MSE) with our proposed method (New B Lasso CQ Reg). This value is 5.077. 

Table 4 also indicates the results of (MSE) for the rq and 𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝑵𝑵 methods. The (MSE) calculated by our 

proposed method (New B Lasso CQ Reg) is much smaller than the (MSE) calculated by rq and  

𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝑵𝑵 methods. This indicates that our proposed method performs better than the rq and 𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝑵𝑵 methods 

for all selected quantile levels. 

5. Conclusions and  recommendations 

In this paper, we developed an efficient method (New B Lasso CQ Reg) to estimate model parameters and 

variables selection simultaneously, in composite quantile regression model via scale mixture uniform as prior 

distributions on the parameters, by using Bayesian approach. The simulation study illustrated that our proposed 

method is more efficient than other methods. The benefits of using the New B Lasso CQ Reg method are 

supported by the simulations performed as well as by the analysis of the real data. The results indicate higher 

accuracy in case of the New B Lasso CQ Reg method. So, we recommended, the researchers are working in 

field of  quantile regression using composite quantile regression . Because of , this approach informative for all 

quantile level under study. 

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