new bayesian lasso in tobit quantile regression · model. these methods were extended to tobit...

Revista Română de Statistică - Supliment nr. 6 / 2017 213

New Bayesian Lasso in Tobit Quantile Regression

Fadel Hamid Hadi ALHUSSEINI (fadhelfadhel222@yahoo)

University of Craiova, Romania

ABSTRACT

In this paper, we proposed a new hierarchy in Bayesian lasso through

using scale mixture uniform (SMU) prior parameters in Tobit quantile

regression (Tobit Q Reg) to achieve coeffi cients estimation and variables

selection. SMU is considered a good replacement for scale mixture normal

(SMN) to satisfy variable selection in Bayesian lasso (Tobit Q Reg). The

Gibbs samplings are derived for all posterior distributions. The performance

assessment of the method proposed versus other methods is done through

simulation examples and real data.

Keywords: New Bayesian lasso, MCMC, Tobit Quantile Regression, scale mixture uniform , variable selection. JEL Classifi cation: C11, C51, C52

INTRODUCTION

Tobit quantile regression (Tobit QReq) is good technique to evaluate the relationship among censored response variable and a set of explanatory variables. Since the seminal work of Powell (1986), Tobit QR has become important in many real-world application sciences, such as econometrics, psychology and social science. After Powell, many researchers have developed different methods for improving parameters estimation for Tobit quantile regression, such as: Hahn (1995) - Buchinsky and Hahn (1998) introduced effi cient bootstrap method; Bilias et al. (2000) introduced modifi cation

for improving bootstrap method; Chernozhukov and Hansen (2008), etc..

Some Tobit QR models have many explanatory variables. These variables

have different relationship with the censored response variable, while some

explanatory variables do not have importance in this model. Therefore,

weak explanatory variables have no justifi cation for their existence in the

model. The diagnosis of the weak independent variables is quite challenging.

Variable selection is a good tool for overcoming this problem. It has a high

quality for building the regression models because this procedure has high

ability for choosing good explanatory variables for regression models and

excluding bad explanatory variables from these models. The development

stages of the variable selection technique are as follows: fi rst AIC appeared

Romanian Statistical Review - Supplement nr. 6 / 2017214

(Akaike,1973) and then BIC (Schwarz,1978). All these traditional methods are poor and have many drawbacks, such as long time to achieve variables selection, (Breiman,1996). In addition, when the number of possible models is large, there is no guarantee that the penalty parameter for model complexity is suitable for comparing subsets with different dimensions. Recently, the researchers have presented innovative modern methods for achieving variable selection in regression models. These methods have good features and do not need long time for variable selection, as it is done automatically. Some of these methods are: lasso (least absolute shrinkage and selection operator) (Tibshirani, 1996), SCAD (Fan and Li, 2001) and elastic net method (Zou and Hastie, 2005). Park and Casella, 2008 presented Bayesian Lasso in traditional regression model. These methods were extended to Tobit quantile regression. instance, (Alhamzawi, 2013), who proposed adaptive lasso in Tobit quantile regression by using Bayesian technique. Alhamzawi and Yu, 2014, suggested a Bayesian technique for coeffi cient estimation in (TobitQReg) model utilizing g-prior

distribution with ridge parameter. Alhamzawi, 2014 proposed a Bayesian

elastic net penalty in (Tobit QReg). Most methods in the fi eld of penalized

Bayesian Tobit quantile regression used the scale mixture of normal

distribution (SMN) before achieving Bayesian Lasso in Regression models.

Mallick and Yi, 2014 provided a new technique for achieving Bayesian lasso

in traditional regression model by scale mixture of uniform formulation

of the Laplace density. We propose a new Bayesian lasso in Tobit quantile

regression through using scale mixture of the uniform distribution (SMU),

from production posterior distributions, we will build tractable and effi cient

algorithm (MCMC (Markov Chain Monte Carlo)). The paper is organized as

follows: section 1 includes a brief review of Tobit quantile regression; section

2 covers the methodology of the new Bayesian lasso Tobit quantile regression,

section 3 includes the evaluation of the behavior of the proposed method by

simulation scenarios, section 4 presents the performance of the proposed

method by real data and section 5 reveals the conclusion of the paper with a

brief discussion.

1. BRIEF REVIEW OF THE TOBIT QUANTILE REGRESSION

Employing the optimum model is a challenging task, but for each

dataset, there is an optimal model. In case of the Tobit regression model, there

is an adaptation with left censored data. Since the seminal work of James

Tobin, 1958, this has model become utilized in all knowledge sciences, such

as economic phenomena, psychology and medicine. This model gives good

estimation when achieving the normal assumptions or when the datasets


are empty from outliers, and it becomes useless when one of the normal assumptions is broken or when outliers exist in the dataset. For overcoming this problem, Tobit quantile regression is employed, which can estimate its coeffi cient without the normal assumption. This model was introduced by

Powell, 1986 and is defi ned by the formula below:

is latent unobserved response variable, ( are intercept and vector unknown

(1)

where

is latent unobserved response variable, ( is latent unobserved response variable, (

is latent unobserved response variable, (,

is latent unobserved response variable, ( are intercept and

vector unknown parameters of Tobit quantile regression respectively, and

parameters of Tobit quantile regression respectively, and is quantile level belonging to the

is censored response variable at censored point (

is quantile level belonging to the open interval (0,1) as open interval (0,1) as , is censored response variFor estimating the parameters of Tobit quan

, is

censored response variable at censored point (C).

For estimating the parameters of Tobit quantile regression, we

minimized the following loss function:

[2] (2)

In Tobit quantile regression, the censored point (C) is equal zero.

Therefore, the loss function (2) takes the following formula:

is called check (loss) function of Koenker and Bassett (1978) at a quantile

(3)

where where is called check (loss) function of Therefore, the eqn [3] can be rewritten as below:

is called check (loss) function of Koenker and Bassett

(1978) at a quantile Koenker and Bassett (1978) at a quantile .. Therefore, the eqn [3] can be rewritten as below:

Therefore, the eqn [3] can be rewritten as below:

(4)

The coeffi cients estimation (Tobit QR) is performed by minimization

of the eqn [3]. The eqn [3] is, however, not differentiable at 0, so there is

not an exact form of the solution for the parameters (Koenker, 2005). The

minimization of eqn [3] can be resolved by a linear programming algorithm

(Koenker and D’Orey, 1987). Although asymptotic properties for TobitQR are

well studied and many algorithms are proposed, most of these algorithms are

ineffi cient, when the amount of censored data is high. Currently, a possible

estimation of coeffi cients (Tobit QR) by (crq) function, exists in package

(quantreg) (Koenker, 2011).


2. METHODOLOGY OF NEW BAYESIAN LASSO TOBIT

QUANTILE REGRESSION

The Bayesian approach has excellent features for parameters estimation, such as it has the ability to estimate exact inference even in the case of small sample size. It can also be updated to model parameters by prior distribution. The Bayesian approach is considered one method for coeffi cient

estimation in regression models, where it depends on likelihood function for

random error term and prior distribution for model parameters, as following:

(5)

2.1. Likelihood Function of Tobit Quantile Regression

Koenker (2005) details a number of algorithms used to estimation.

Tobit QR was ineffi cient when the response variable dataset has a high

amount of censored data. Instance linear programming algorithm (Koenker

and D’Orey, 1987) and others. Recently, Yu and Stander, 2007 have proposed

the Bayesian approach for estimating (Tobit QR) even with high amount of

censored data. They were inspired by the suggestion of Konker and Machado

(1999) and Yu and Moyeed (2001). These researchers observed convergence

between loss function (3) and skew-Laplace distribution (SLD) (asymmetric

Laplace distribution). Therefore, the random error term ui distributed as SLD

with probability density function (pdf), takes the following formula:

(6)

If and then, the probability density function (pdf) to u is:

(7)

where where belong to interval (0,1) and

joint distribution of

belong to interval (0,1) and belong to interval (0,1) and is the check function defined as in eqn [4]. The is the

check function defi ned as in eqn [4]. The joint distribution of belong to interval (0,1) and is the check function defined as in eqn [4]. The

joint distribution of is: is:

where the likelihood function to probability density function (pdf) of (SLD) with scale

(8)


where the likelihood function to probability density function (pdf) of (SLD) with scale parameter equal one is:

From known, minimizing the eqn [3] is equivalent to maximizing the likelihood

(9)

From known, minimizing the eqn [3] is equivalent to maximizing the likelihood function of the eqn [9]. By using SLD directly, leads to hard computations, therefore Kozumi and Kobayashi, 2011, indicated that SLD can be reformulated as function scale mixture normal distribution. The likelihood function of eqn [9] is possible to be rewritten as the following equation, according to suggestion of Kozumi and Kobayashi, 2011):

is distributed experiential distribution with rate parameter

(10)

where zi is distributed experiential distribution with rate parameter stribution with rate parameter . The eqn [10] is

of posterior distributions for coefficient . The eqn [10] is an important part for building Gibbs samplers of

posterior distributions for coeffi cient estimation to Tobit quantile regression.

2.2. Bayesian Hierarchical of Prior Distributions

Tibshirani, 1996, implement the Bayesian lasso in traditional

regression model by assigning Laplace priors for regression coeffi cients. The

Laplace distribution has the probability density function with

assigning Laplace priors for regression coef

probability density function with as below: as below:

is the shrinkage parameter and .

(11)

where

is the shrinkage parameter and

is the shrinkage parameter and is the shrinkage parameter and .More researchers discussed Bayesian regular

.

More researchers discussed Bayesian regularize Tobit quantile

regression. For instance, Yue and Hong, 2012, proposed Bayesian Tobit

QR with the group Lasso penalty, Alhamzawi, 2013, proposed the adaptive

lasso in Tobit QR by using Bayesian technique, Alhamzawi, 2014 proposed

a Bayesian elastic net penalty in Tobit QReg. Where most last methods using

another picture from Laplace prior is scale mixture normal (SMN), according

to Andrews and Mallows (1974).


Hence, Laplace prior can be rewritten as a function from two parts. The fi rst part can be assigned to prior distribution for be assigned to prior distribution for ,which distributes normally with mean zero and variance , which distributes

normally with mean zero and variance (( ) as follows: ) as follows:

unknown variance is .

(12)

where unknown variance is .

The exponential prior for takes the form of:

(13)

In this paper, we proposed a new Bayesian lasso Tobit QR, through

using another formula of the Laplace prior which is scale mixture uniform. It

is employed in the Bayesian lasso approach in classical regression model by

Mallick and Yi, 2014. In this case, the Laplace prior of coeffi cients in eqn

[11] is written as follows:

(14)

Eqn [14] can be rewritten as a function from two parts. The fi rst part

is assigned to prior uniform distribution for uniform distribution for ,

parameter (2) and scale parameter (

, and the second part belongs to

Gamma distribution with shape parameter (2) and scale parameter (parameter (2) and scale parameter ( ), where

parameter is necessary for coefficient shrinkage. The prio

), where

parameter (2) and scale parameter ( ), where

parameter is necessary for coefficient shrinkage. The prio

has Gamma prior with parameters (a, b). This parameter is necessary for

coeffi cient shrinkage. The prior distribution of r distribution of is assigned to is assigned to uniform

prior. a, b which are fi xed hyperparameters.

From the above data, our Bayesian hierarchical Tobit quantile

regression model can be summarized as follows:

,

(15)


2.3. The full Conditional Posterior Distributions Inference

From eqn [10] and the set of eqn [15], we will obtain the condition posterior distribution. Therefore, building the Gibbs sampler depends on the following: • The full posterior distribution of The full posterior distribution of is obtained from the formula is obtained from the formula below:

• The full posterior distribution of The full posterior distribution of is normal distribution with mean is normal distribution with

mean is normal distribution with mean and the and the variance variance

• The full posterior distribution of zi is inverse Gaussian as the following

formula

The full posterior distribution of is inverse Gaussian as the following

formula and

• The full posterior distribution of The full posterior distribution of is truncated normal taken the following picture: is truncated normal taken the following picture:The full posterior distribution of is truncated normal taken the following picture:

where and

• The full posterior distribution of sj is left-truncated exponential as the following formula with rate parameter formula with rate parameter : .

executable by using inversion method (Mallick and Yi,2014) Update sj executable by using inversion method (Mallick and Yi,2014) a. Updating sj* from exponential distribution with rate parameter of (parameter (2) and scale parameter ( ), where

parameter is necessary for coefficient shrinkage. The prio)

b. Dataset Dataset = • The full posterior distribution of tuning parameter (parameter (2) and scale parameter ( ), where

parameter is necessary for coefficient shrinkage. The prio)

is Gamma distribution with rate parameter parameter and scale parameter and scale parameter tion of tuning parameter ( is Gamma distribution with rate and scale parameter ,

3. SIMULATION STUDIES

In this section, the performance of our method is assessed in a simulation study in which the method for a new Bayesian lasso in Tobit quantile regression model (Tobit Q Reg) is compared to two classical Tobit quantile regression method documented by Powell, 1986, through using function (crq) in R within package quantreg (Koenkers, 2011). The Bayesian elastic net penalty in (Tobit Q Reg) has the value proposed by Alhamzawi, 2014. For comparison, we used two criteria:


• The median of mean absolute deviations summarized (MMAD), where, MMAD =

The median of mean absolute deviations summarized (MMAD), where,

. . • The standard deviation of the MADS; in this case, we used three quantile levels, as follows: lower quantile at

The standard deviation of the MADS; in this case, we used three quantile levels, as follows: lower quantile at , middle quantile at

; the data generated by depend on true , middle quantile at MADS; in this case, we used three quantile levels, as

, middle quantile at and higher quantile at model is describe by following model:

and higher quantile at ; the data generated by depend on true ; the data generated by depend on true model is describe by following model:

.The error term . is generated from three different distributions: a normal The error term The error term . is generated from three di

distribution with mean one and vari is generated from three different

distributions: a normal distribution with mean one and variance one, stander Laplace distribution

distribution with mean one and vari, distribution with four

The number of simulations was one hundred for each case. Our algorithm was run distribution with four degrees

of freedom. The number of simulations was one hundred for each case. Our algorithm was run 13000 iterations. The fi rst 3000 were ruled out as burn in.

To assess the performance of our method, we used three approaches:

The fi rst approach

The simulated studies are performed with very sparse case with adding

the intercept term to true parameters

The simulated studies are performed with ve

term to true parameters . Therefore, the true model becomes as follow: . Therefore,

the true model becomes as follow:

We simulated the explanatory variables We simulated the explanatory variables ( from a

, where is the mean vector and from a multivariate Gaussian

distribution

We simulated the explanatory variables (

multivariate Gaussian distribution , where

.

, where , where is the mean vector is the mean vector

from a

is the mean vector and and

from a

and is is

covariance matrix with

multivariate Gaussian distribution

covariance matrix with .

Figure 1. shows a clear picture of the median

.

Figure 1 shows a clear picture of the median of mean absolute

deviations and standard deviation, for the three methods under consideration.

When comparing our proposed method (New B Tobit QReg) with two

other methods (Banet and crq), we fi nd that our proposed method has better

performance than the other methods. The proposed method has MMAD

smaller than Banet and crq across different error distributions and different

quantile levels. This result supports the performance of the proposed method

comparison by two other method. The proposed method has also smaller

standard deviation (SD) through different error distributions and different

quantile levels than the other two methods. Only in the case of crq method, the

values were better than for the proposed method at lower quantile in chi-square.


Bar charts summarizing the values of MMADs and the standard

deviations MADS (SD) for our method and two other methods by using

three error distributions in the fi rst approach through three quantiles

levels (30, 60, 90)

Figure 1

The second approach

In this approach the simulated studies were with dense case and adding the intercept to true parameters In this approach the simulated studies were with dense case and adding the intercept to

true parameters . Therefore, the true model takes the following formula:

. Therefore, the true model takes the following formula:


The explanatory variables are simulate ( according to

The explanatory variables are simulate The explanatory variables are simulate ( according to

, where is the mean vector according to a multivariate Gaussian

distribution The explanatory variables are simulate (

a multivariate Gaussian distribution , where .

where The explanatory variables are simulate (

is the mean vector according to

is the mean vector and is is cov-matrix with

a multivariate Gaussian distribution cov-matrix with .

From Figure 2. we can clearly see that .

From Figure 2. we can clearly see that our proposed method (New B Tobit QReg) performs better than the other two methods (BAnet and crq). This inference is obvious from the value of the median of mean absolute deviations(MMAD) and standard deviation (SD), for the three methods under consideration. The proposed method (New B Tobit QReg) has the smallest MMAD in comparison to the other two methods (BAnet and crq). Generally, our proposed method has better performance than the other two methods across different error distributions and different quantile levels.



deviations MADS (SD) for our method and two other methods by using

three error distributions in the second approach through three quantiles

levels (30, 60, 90)

Figure 2

The third approach

The simulated studies are done with group structures and adding zero as intercept to true parameters

The simulated studies are done with group structures and adding zertrue parameters .Therefore, the true model takes the below formula:

Therefore, the true model takes the below formula:


takes the below formula:

where the explanatory variables are simulated. where the explanatory variables are simulated. according to a

Gaussian distribution ,where is mean vector according

to a multivariate Gaussian distribution Gaussian distribution ,where,where ,where is mean vector is mean vector is mean vector and is cov-matrix with is cov-matrix with

Gaussian distribution

From Figure 3. we can see clearly that

. From Figure 3. we can see clearly that our proposed method (New B Tobit QReg) generally performs better than two other methods (BAnet and crq) for all quantile levels and all different error distributions. This inference is obvious from the value of the median of mean absolute deviations (MMAD) and standard deviation (SD), for three methods under consideration. The results show that the proposed method (New B Tobit QReg) has the smallest MMAD and SD comparisons with two other methods (BAnet and crq).



deviations MADS(SD) for our method and other two methods by using

three error distributions in the third approach through three quantiles

levels (30, 60, 90)

Figure 3

For evaluation of our proposed method (New B Tobit QReg) against two other methods (Banet and crq) we will use another criterion, parameters estimations in direct way.


True parameters values and posterior means to simulation data in fi rst

approach via three error distribution and three quantile levels

Table 1

Errordistribution

Quantilelevels

Method

True parameters

0 1 0 0 0 0 0 0 1

crq 1.401 1.45 0.501 -1.664 1.541 -1.641 -0.509 -0.081 1.388 BAnet 0.392 0.016 -0.108 -0.679 0.094 0.119 0.08 -0.763 1.585 New B

Tobit QReg 0.308 1.023 0.021 0.016 0.017 0.039 0.186 -0.029 1.038

crq -0.023 1.469 1.138 -0.923 -0.74 1.543 1.55 -2.089 3.529 BAnet -0.037 1.561 -0.17 -0.328 0.409 -0.274 1.75 -0.593 1.722 New B Tobit QReg

0.008 1.056 -0.018 -0.062 0.235 0.045 0.768 0.017 0.701

crq 1.558 1.572 -0.679 -2.387 -0.57 4.683 -1.983 -0.805 6.374 BAnet 1.019 0.612 -0.236 -0.207 -1.52 2.307 0.079 -0.022 1.847 New B Tobit QReg

0.492 0.461 -0.023 -0.148 -0.53 0.991 0.365 0.006 0.958

True parameters

0 1 0 0 0 0 0 0 1

crq -0.534 3.086 -0.247 -0.493 0.001 -0.605 0.372 0.78 1.518 BAnet -0.074 1.788 -0.178 -0.533 0.257 -0.508 0.234 0.347 1.439 New B Tobit QReg

0.712 1.045 0.012 -0.507 0.061 -0.024 0.327 0.049 1.061

crq 0.481 1.691 -0.779 -0.696 0.876 -1.002 0.727 1.52 0.629 BAnet 0.269 1.923 -0.755 0.364 -0.43 -0.373 0.865 0.145 1.144 New B Tobit QReg

0.09 0.989 0.224 0.125 -0.12 0.021 339 0.109 0.504

crq -0.835 5.009 -0.145 -4.334 -0.71 -0.499 1.937 0.079 4.507 BAnet 0.711 2.022 -0.59 -0.683 1.036 -0.189 -0.351 -0.182 2.75 New B Tobit QReg

0.934 1.125 0.191 -0.292 0.663 0.139 0.118 0.032 0.985

True parameters

0 1 0 0 0 0 0 0 1

crq 0.407 1.985 0.076 0.224 -0.11 -0.611 -0.371 -0.163 2.097 BAnet 0.395 0.987 0.105 0.672 -0.57 -0.432 -0.252 -0.088 1.834 New B Tobit QReg

0.314 1.014 0.203 0.035 0.101 -0.218 -0.212 0.029 1.578

crq -0.204 2.217 -0.839 2.13 -1.34 -0.331 -0.169 1.546 3.001 BAnet -0.612 2.147 -0.431 1.027 -0.39 -0.394 0.072 -0.016 1.49 New B Tobit QReg

-0.116 1.103 0.358 0.283 0.133 0.009 0.097 0.194 0.791

crq -0.315 5.774 2.464 -3.325 1.307 1.125 0. 252 -3.557 2.567 BAnet -0.147 2.289 0.539 -1.329 1.686 0.367 0.613 -0.621 1.315 New B Tobit QReg

0.125 1.347 0.323 -0.443 0.726 0.222 0.239 -0.021 0.496

From Table 1. we can see that the parameters estimation by our proposed method (New B Tobit QReg) was very closed to true parameters. The estimations of parameters by two other methods (BAnet and crq) were


fare to the true parameters approximately. From these results, we conclude that our proposed method (New B Tobit QReg) has performed better than BAnet and crq methods.

4. REAL DATA

To show the performance of our proposed method (New B Tobit QReg) and compare it with Banet method, we use Extramarital Affairs data which is included in the AER package from R. This data was presented by Fair in 1978 and has been used in at least a hundred papers (Ji et al., 2012, Alhamzawi, 2014, and others). The response variable represents the number of times extramarital sexual encounters occurred during the past year (affairs). It is censored from left side and the dataset consists of n=601 observations, out of which 451 observations are censored, and rest of the observations uncensored. The eight covariates are gender (1 for female and 2 for male), gender, age, number of years married, children (2 for existence of children in the marriage and 1 without children), religiousness (scale from 1 to 5), level of education, how much prestige their occupation (scale from 1 to 7) and rating the happiness in their marriage (scale from 1 to 5). As like in section 3, where our algorithm is run 13000 iterations and the fi rst 3000 iterations are removed,

Table 2. Shows the parameters estimation for our proposed method (New B

Tobit QReg), and (BAnet) method across four different quantiles as follows:

Estimation of model parameters through two methods across four

different quantiles

Table 2Table 2

New B Tobit QReg BAnetVariables Intercept 0.027 0.071 0.264 0.591 5.675 4.176 10.522 -3.337 Gender 0.019 0.012 0.033 0.016 0.063 0.735 0.654 3.524 Age -0.001 0.001 0.003 0.019 -0.362 -0.100 -0.101 0.490 Years married 0.004 0.006 0.033 0.133 0.015 0.176 0.633 0.506 Children 0.035 0.055 0.100 0.073 -1.053 -1.838 0.787 1.962 Religiousness -0.004 -0.007 -0.062 -0.158 -0.562 -0.458 -1.086 -0.066 Education 0.013 0.028 0.078 0.106 -0.321 -0.037 0.013 0.899 Occupation 0.001 0.005 0.022 0.052 0.022 -0.178 0.345 -0.207 Rating -0.008 -0.019 -0.095 -0.209 -1.075 -0.779 -1.064 0.316

Table 2. shows the parameter estimates through two methods (New B

Tobit QReg and BAnet), which were used to calculate the mean square error

(MSE) to our proposed method (New B Tobit QReg) as in the results from

Table 3.


Mean square errors for our proposed method (NewBQ Reg) and

(BAnet) method

Table 3

Table 3

MethodsNewBQ Reg 12.19164 11.48281 10.47575 12.83809

BAnet 433.4153 59.47696 70.83432 1559.169

Table 2. shows the values of MSE for both methods across four quantile levels. The mean square error to our proposed method (NewBQ Reg) is much smaller compared to BAnet method for each quantile levels. The results refl ect that our method is more effi cient than BAnet. Based on the results of criteria in both simulation studies and real data, our proposed method (NewBQ Reg) appears to have a good performance.

5.CONCLUSION

Our proposed method (NewBQ Reg) depends on scale mixture uniform as prior distribution. Therefore, it is considered a new approach in Bayesian Tobit quantile regression. The results in both simulation study and real data recorded superior results when employing our proposed method (NewBQ Reg) compared to other methods through different quantile levels. Thus, it can be considered a valid way for coeffi cient estimation and variable selection in Tobit quantile regression. Further development on our proposed method (NewBQ Reg) are possible in several trends, for instance Bayesian composite Tobit quantile regression with scale mixture uniform.

References 1. Akaike, H., 1973, “Information theory and an extension of the maximum likelihood

principle”, Petrov, B.N. and Caski, F. eds., Akademiai Kiado, Budapest, 1973, pp.267-281.

2. Alhamzawi, R., 2013, “Tobit quantile regression with adaptive lasso penalty”, The 4th International Scientifi c Conference of Arab Statistics 450 ISSN, pp. 1681-6870)

3. Alhamzawi, R., 2014, “Bayesian elastic net tobit quantile regression”, Communications in Statistics - Simulation and Computation (to appear).

4. Alhamzawi, R., Yu K., 2014, “Bayesian tobit quantile regression using g-prior distribution with ridge parameter”, Journal of Statistical Computation and Simulation (ahead-of-print), 1-16.

5. Bilias, Y., Chen S., Ying Z., 2000, “Simple resampling methods for censored regression quantiles, Journal of Econometrics, 68, pp. 303-338.

6. Buchinsky, M., Hahn J., 1998, “An alternative estimator for censored quantile regression”, Journal of Econometrica, 66, pp. 653-671.


7. Breiman, L., 1996, “Heuristics of instability and stabilization in model selection”, Journal of the Annals of Statistics, 24 (6), pp. 2350-2383.

8. Chernozhukov, V., Hansen C., 2008, “Instrumental variable quantile regression: A robust inference approach”, Journal of Econometrics 142 (1), pp. 379-398.

9. Fair, R., 1978, “Theory of extramarital affairs”, Journal of Political Economy, 86, pp. 45-61.

10. Fan, J., Li R. Z., 2001, “Variable selection via nonconcave penalized likelihood and its oracle properties”, Journal of the American Statistical Association, 96, pp. 1348-1360. 5.

11. Hahn, J., 1995, “Bootstrapping quantile regression estimators”, Econometric Theory, 11, pp. 105-121.

12. Ji, Y., Lin N., Zhang B., 2012, “Model selection in binary and Tobit quantile regression using the Gibbs sampler”, Computational Statistics & Data Analysis 56, pp. 827-839.

13. Kozumi, H., Kobayashi G., 2011, “Gibbs sampling methods for Bayesian quantile regression”, Journal of Statistical Computation and Simulation, 81, pp. 1565-1578.

14. Koenker, R., Bassett G. J., 1978, “Regression quantiles”, Econometrica, 46, pp. 33-50.

15. Koenker, R., D’Orey V., 1987, “Algorithm AS 229: Computing regression quantiles”, Journal of the Royal Statistical Society: Series C (Applied Statistics), 36, pp. 383-393.

16. Koenker, R., Machado J. A. F., 1999, “Goodness of _t and related inference processes for quantile regression”, Journal of the American Statistical Association, 94, pp. 1296-1310.

17. Koenker, R., 2005, “Quantile Regression”, Cambridge Books, Cambridge University Press.

18. Koenker, R., 2011, “quantreg: Quantile regression. R package version 5.05”. 19. Mallick, H., Yi, N., 2014, “A new Bayesian lasso”, S statistics and its interface,

7(4), pp. 571-582. 20. Powell, J., 1986, “Censored regression quantiles”, Journal of Econometrics, 32,

pp. 143-155. 21. Park, T., Casella, G., 2008, “The Bayesian lasso”, Journal of the American

Statistical Association, 103(482), pp. 681- 686. 2. 22. Tibshirani, R., 1996, “Regression shrinkage and selection via the lasso”, Journal

of the Royal Statistical Society. Series B (Methodological), pp. 267-288. 2 23. Tobin, J., 1958, “Estimation of relationships for limited dependent variables”,

Econometrica, 26, pp. 24-36. 24. Zou, H., 2006, “The adaptive lasso and its oracle properties”, Journal of the

American Statistical Association, 101(476), pp. 1418-1429. 2. 25. Yu, K. & R. Moyeed A. (2001). “Bayesian quantile regression”, Statistics &

Probability Letters, 54(4), pp. 437-447.2. 26. Yu, K., Stander J., 2007, “Bayesian analysis of a Tobit quantile regression model”,

Journal of Econometrics 137, pp. 260-276.

new bayesian lasso in tobit quantile regression · model. these methods were extended to tobit...

Documents