single parameter defines both the quantile and the skewness of the distribution. Besides, the mode of the distribution is always fixed at the location parameter value for all quantiles. To overcome these drawbacks, Yan and Kottas (2017) proposed the probability density function ( pdf ) of the GAL distribution by introducing a shape parameter into the mean of the normal kernel in the AL mixture representation. The GAL distribution uses different parameters for quantile and skewness, and thus adds the much needed flexibility required for Bayesian quantile regression. Yan and Kottas (2017) utilized the GAL pdf and proposed algorithms for Bayesian quantile estimation of linear models, Tobit models and regularized quantile regression. 

In this paper, we present a derivation of the GAL pdf from the mixture representation and introduce the cumulative distribution function ( cdf ) and the moment generating function ( mgf ) of the GAL distribution along with their derivations. The hierarchical representation of the GAL density and the GAL cdf are utilized to introduce a Bayesian estimation algo- rithm for the FBQROR model. Estimation of ordinal models, unlike linear models, is more intricate since there are identification restrictions and sampling of cut-points has to satisfy the ordering constraints (Jeliazkov et al., 2008; Rahman, 2016). Moreover, through careful transformation of the mixture variables and joint sampling of the scale and shape parameters, we are able to achieve low autocorrelation in our MCMC draws. This is in sharp contrast to Yan and Kottas (2017), where they find high autocorrelation in the MCMC draws and resort to thinning, which is always inefficient and rarely useful (Link and Eaton, 2012). Our estimation scheme can therefore improve the algorithm for Bayesian quantile regression in linear and Tobit models, and regularized regression as presented in Yan and Kottas (2017). We illustrate the proposed methodology in two simulation studies wherein the errors are generated from a symmetric (logistic) distribution and an asymmetric (chi-square) distribu- tion. Model comparison reveals that the FBQROR models can provide a better model fit compared to that obtained from Bayesian quantile regression in ordinal (BQROR) models assuming an AL distribution (Rahman, 2016). Finally, we implement our FBQROR model 

