Appendix A. The GAL Distribution This appendix derives the pdf of the GAL distribution from the mixture representation (Yan and Kottas, 2017), introduces the cdf and mgf of the GAL distribution. The mgf is also utilized to derive the mean, variance and skewness of the GAL distribution. Theorem 1 : Suppose Y ∼ GAL ( µ, σ, p, α ) and has the pdf given by equation (2), then Y has the following hierarchical representation, Y = µ + ασS + σA ( p ) W + σ [ B ( p ) W ] 1 2 U , where all the notations are as in Section 2. Proof : Using the mixture representation we can write the pdf of Y as, where a = σA ( p ) = σ (1 − 2 p ) p (1 − p ) , b = µ + σαs , and c = σ $^{2}$B ( p ) = 2 σ 2 p (1 − p $_{)}$. We let P denote the second integral and integrate with respect to w as follows, where third line makes the substitutions γ = a $^{2}$+2 c c = 1 2 p (1 − p $_{)}$, µ 2 = γc ( y − b ) 2 = σ 2 p $^{2}$(1 − p ) $^{2}$( y − b ) 2 , 27 P = ∫ ∞ 0 1 √$_{2}$$_{πw}$ exp { − 1 2 [ ( a 2 + 2 c ) w 2 + ( y − b ) 2 − 2 a ( y − b ) w ) cw ]} dw = exp { a ( y − b ) c } ∫ ∞ 0 1 √$_{2}$$_{πw}$ exp { − 1 2 [ ( a 2 + 2 c ) c w + ( y − b ) 2 cw ]} dw = exp { a ( y − b ) c } ∫ ∞ 0 √ γ 2 πw √ 1 γ exp { − 1 2 [ γw + γ µ $^{2}$w − 2 γ µ + 2 γ µ ]} dw = 1 √ γ exp { − γ µ } exp { a ( y − b ) c } ∫ ∞ 0 √ γ 2 πw exp { − γ 2 ( (1 − µw ) 2 µ $^{2}$w )} dw ︸ ︷︷ ︸ integrates to 1 = 1 √ γ exp { − γ µ } exp { a ( y − b ) c } (for γ, µ > 0) , (A.2) f ( y | θ ) = ∫ R + ∫ R + N ( y | µ + σαs + σA ( p ) w, σ $^{2}$B ( p ) w ) exp( w | 1) N $^{+}$( s | 0 , 1) dw ds = R + 1 √ σ $^{2}$B ( p ) ∫ ∞ 0 [ 1 √$_{2}$$_{πw}$ exp { − 1 2 [ ( y − b − aw ) 2 cw ] − w } dw ] N $^{+}$( s | 0 , 1) ds (A.1) Appendix A.1. Probability Density Function