Secondly, we integrate the expression for M$_{2}$ as follows, M$_{2}$ = − κ ∫ ∞ µ exp { ty } Φ( − αp$_{α}$$_{−}$ ) exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } dy = − κ Φ( − αp$_{α}$$_{−}$ ) exp { p$_{α}$$_{−}$µ σ + 1 2 α $^{2}$p2 α$_{−}$ } ∫ ∞ µ exp { − ( p$_{α}$$_{−}$ − σt ) σ y } dy = − κ Φ( − αp$_{α}$$_{−}$ ) exp { p$_{α}$$_{−}$µ σ + 1 2 α $^{2}$p2 α$_{−}$ } × ( − 1)( − σ ) ( p$_{α}$$_{−}$ σt ) exp { − ( p$_{α}$$_{−}$ − σt ) σ µ } = − κσ exp { µt + 1 $_{2}$α$^{2}$p 2 α$_{−}$ } ( p$_{α}$$_{−}$ − σt ) Φ( − αp$_{α}$$_{−}$ ) (limit exists only if t < p/σ ) . (A.22) The integral M$_{3}$ is evaluated by substituting z = αp$_{α}$$_{+}$ − y ∗ α . Thereafter the integration is analogous to M$_{1}$ with the existence condition t > ( p − 1) /α and yields, M$_{3}$ = κσ exp { µt + 1 $_{2}$α$^{2}$p 2 α + } ( p$_{α}$$_{+}$ − σt ) Φ( αp$_{α}$$_{+}$ ) − κσ exp { µt + 1 $_{2}$α$^{2}$σ$^{2}$t 2 } ( p$_{α}$$_{+}$ − σt ) Φ( ασt ) . (A.23) M$_{4}$ = − κσ exp { µt + 1 $_{2}$α$^{2}$p 2 α + } ( p$_{α}$$_{+}$ − σt ) Φ( αp$_{α}$$_{+}$ ) . (A.24) M$_{Y}$ ( t ) = 2 p (1 − p ) [ ( p$_{α}$$_{+}$ − p$_{α}$$_{−}$ ) ( p$_{α}$$_{−}$ − σt )( p$_{α}$$_{+}$ − σt ) ] exp { µt + 1 2 α $^{2}$σ$^{2}$t2 } Φ( ασt ) . (A.25) M$_{5}$ = κ ∫ µ $_{−∞}$exp { ty } Φ ( y ∗ α − αp$_{α}$$_{−}$ ) exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 ( αp$_{α}$$_{−}$ ) 2 } dy, The integral M$_{4}$ can be directly evaluated and closely follows the steps in the integration of M$_{2}$ with the resulting expression, where the limits of integration exists only for t > ( p − 1) /σ . To obtain the mgf of the GAL distribution, we substitute the values of κ and sum the four equations (A.21), (A.22), (A.23) and (A.24). This yields, Case (ii) : When α < 0 and y ∗ ≤ 0 (i.e., y ≤ µ ), we have the following three components of the mgf , 35