content_image/318240.jpg content_image/318239.jpg content_image/318238.jpg
2011). So, we reparameterize and write the model as, 

<img src='content_image/318240.jpg'>

where h$_{i}$ = σs$_{i}$ and ν$_{i}$ = σw$_{i}$ , which in turn imply that h ∼ N $^{+}$(0 , σ $^{2}$) and ν ∼ E ( σ ). Both reformulations are necessary for computational efficiency of the MCMC algorithm. Also, note that the first reparameterization was not utilized in Yan and Kottas (2017) and hence our approach can improve the Bayesian quantile estimation of linear models, Tobit models and lasso regularized quantile regression presented in Yan and Kottas (2017). 

Ordinal models present two additional challenges: location and scale restrictions for identification of the parameters and ordering constraints in sampling of cut-points ξ (see Jeliazkov et al., 2008; Rahman, 2016). In the FBQROR model, both location and scale re- strictions are enforced by fixing two cut-points since the variance of a GAL distribution is not fixed due to its dependence on α even if we set σ = 1, as shown in Theorem 4 in Appendix A. The ordering constraint is resolved by using the following logarithmic transformation, 

<img src='content_image/318239.jpg'>

The original cut-points can then be obtained using equation (12) by one-to-one mapping between δ = ( δ$_{1}$, · · · , δ$_{J}$$_{−}$$_{3}$ ) ′ and ξ = ( ξ$_{3}$, · · · , ξ$_{J}$$_{−}$$_{1}$ ) $^{′}$, where ξ$_{2}$ is fixed at some constant c , and recall that ξ$_{0}$ = −∞ , ξ$_{1}$ = 0 and ξ$_{J}$ = ∞ . 

We next employ the Bayes’ theorem and derive the joint posterior density as propor- tional to the product of the likelihood and prior distributions. We employ standard prior distributions as follows, 

<img src='content_image/318238.jpg'>

