bution that has the following pdf : where η = ( µ, σ, γ ), p ≡ p ( γ, p$_{0}$ ) = I ( γ < 0) + [ p$_{0}$ − I ( γ < 0)] /g ( γ ), p$_{γ}$$_{+}$ = p − I ( γ > 0) and p$_{γ}$$_{−}$ = p − I ( γ < 0). The function g ( γ ) = 2Φ( −| γ | ) exp( γ $^{2}$/ 2) and γ ∈ ( L, U ), where L is the negative square root of g ( γ ) = 1 − p$_{0}$ and U is the positive square root of g ( γ ) = p$_{0}$ (Yan and Kottas, 2017). The term “quantile-fixed” suggests that integration of GAL pdf (6) to the upper limit µ equals p$_{0}$ , so for regression purpose we can fix the quantile. The cdf for the quantile-fixed GAL density (6) can be analogously derived as in Appendix A.2 to yield the following expression, The quantile-fixed cdf (7) is required for constructing the likelihood of the FBQROR model and plays a critical role in the MCMC sampling of the scale parameter, shape parameter and cut-points/thresholds. To better discern the GAL distribution, Figure 1 presents a graphical comparison of the quantile-fixed GAL and AL pdf ’s for three different quantiles. We observe that the GAL distribution, unlike the AL distribution, allows the mode to vary rather than being fixed at µ = 0 at all quantiles. Besides, the GAL distribution can be positive or negatively skewed at all quantiles depending on the value of γ . For example, at the median p$_{0}$ = 0 . 50 the GAL distribution is positively skewed for γ < 0 and negatively skewed for γ > 0. Also, the GAL 6 f$_{p}$$_{0}$ ( y | η ) = 2 p (1 − p ) σ   [ Φ ( − y ∗ p$_{γ}$$_{+}$ | γ | + p$_{γ}$$_{−}$ p$_{γ}$$_{+}$ | γ | ) − Φ ( p$_{γ}$$_{−}$ p$_{γ}$$_{+}$ | γ | ) ] exp { − y $^{∗}$p$_{γ}$$_{−}$ + γ 2 2 ( p$_{γ}$$_{−}$ p$_{γ}$$_{+}$ ) 2 } × I ( y ∗ γ > 0 ) + Φ ( −| γ | + y ∗ p$_{γ}$$_{+}$ | γ | I ( y ∗ γ > 0 )) exp { − y $^{∗}$p$_{γ}$$_{+}$ + γ 2 2 }   , (6) F$_{p}$$_{0}$ ( y | η ) =   1 − 2Φ ( y ∗ p$_{γ}$$_{+}$ γ ) + 2 p$_{γ}$$_{+}$ exp { − y $^{∗}$p$_{γ}$$_{−}$ + γ 2 2 ( p$_{γ}$$_{−}$ p$_{γ}$$_{+}$ ) 2 }[ Φ ( − y ∗ p$_{γ}$$_{+}$ | γ | + p$_{γ}$$_{−}$ p$_{γ}$$_{+}$ | γ | ) − Φ ( p$_{γ}$$_{−}$ p$_{γ}$$_{+}$ | γ | ) ]   I ( y ∗ γ > 0 ) + I ( γ < 0) + 2 p$_{γ}$$_{−}$ exp { − y $^{∗}$p$_{γ}$$_{+}$ + γ 2 2 } × Φ ( −| γ | + y ∗ p$_{γ}$$_{+}$ | γ | I ( y ∗ γ > 0 )) .