where θ = ( µ, σ, p, α ), y ∗ = ( y − µ ) /σ , µ is the location parameter, σ is the scale parameter, α is the shape parameter, p$_{α}$$_{+}$ = p − I ( α > 0) and p$_{α}$$_{−}$ = p − I ( α < 0) with p ∈ (0 , 1). The derivation of the GAL pdf from the hierarchical representation is presented in Appendix A.1 and largely follows the notations used in Yan and Kottas (2017). Note that when α = 0, the GAL pdf reduces to the pdf of an AL distribution. We explore the GAL distribution in greater detail and propose the cdf and mgf of the GAL distribution. The cdf denoted by F can be compactly written as, and the mgf denoted by M$_{Y}$ ( t ) has the following expression, Both the cdf and mgf have been derived and presented in Appendix A.2 and Appendix A.3, respectively. In addition, Appendix A.3 utilizes the mgf (4) to derive the mean, variance and skewness of the distribution. These distributional characteristics are extremely important for better understanding of the GAL distribution and for further development of flexible Bayesian quantile regression. However, the GAL density given by equation (2) has the limitation that the parameter p no longer corresponds to the cumulative probability at the quantile for α = 0. Following, Yan and Kottas (2017), we let γ = [ I ( α > 0) − p ] | α | and re-express the mixture representation (1) as follows, where C = [ I ( γ > 0) − p ] − $^{1}$. This re-parametrization yields the quantile-fixed GAL distri- 5 Y = µ + σAW + σC | γ | S + σ [ BW ] 1 2 U, (5) M$_{Y}$ ( t ) = 2 p (1 − p ) [ ( p$_{α}$$_{+}$ − p$_{α}$$_{−}$ ) ( p$_{α}$$_{−}$ − σt )( p$_{α}$$_{+}$ − σt ) ] exp { µt + 1 2 α $^{2}$σ$^{2}$t2 } Φ ( | α | σt ) . (4) F ( y | θ ) = ( 1 − 2Φ ( − y ∗ | α $_{|}$) − 2 p (1 − p ) p$_{α}$$_{−}$ exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ }[ Φ ( y ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ]) I ( y ∗ α > 0 ) + I ( α < 0) − 2 p (1 − p ) p$_{α}$$_{+}$ exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 α $^{2}$p2 α + } × Φ [ αp$_{α}$$_{+}$ − y ∗ α I ( y ∗ α > 0 )] , (3)