Case (i) : When ( α > 0 , y ≤ µ ) or ( α < 0 , y > µ ), the cdf is, F ( y | θ ) =      y ∫ −∞ κ Φ( αp$_{α}$$_{+}$ ) exp { − v $^{∗}$p$_{α}$$_{+}$ + 1 $_{2}$α$^{2}$p 2 α + } dv, if α > 0 , y ≤ µ 1 − ∞ ∫ y κ Φ( αp$_{α}$$_{+}$ ) exp { − v $^{∗}$p$_{α}$$_{+}$ + 1 $_{2}$α$^{2}$p 2 α + } dv, if α < 0 , y > µ =     $^{κ}$$^{Φ(}$$^{αp$_{α}$$_{+}$}$$^{) exp}$ { 1 $_{2}$α$^{2}$p 2 α + } [ exp { − v ∗ p$_{α}$$_{+}$ } − p$_{α}$$_{+}$ /σ ] y −∞ , if α > 0 , y ≤ µ 1 − κ Φ( αp$_{α}$$_{+}$ ) exp { 1 $_{2}$α$^{2}$p 2 α + } [ exp { − v ∗ p$_{α}$$_{+}$ } − p$_{α}$$_{+}$/σ ] ∞ y , if α < 0 , y > µ =     $^{2}$$^{p}$$^{(1}$$^{−}$$^{p}$$^{)}$ − p$_{α}$$_{+}$ Φ( αp$_{α}$$_{+}$ ) exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 $_{2}$α$^{2}$p 2 α + } , if α > 0 , y ≤ µ 1 − 2 p (1 − p ) p$_{α}$$_{+}$ Φ( αp$_{α}$$_{+}$ ) exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 $_{2}$α$^{2}$p 2 α + } , if α < 0 , y > µ (A.10) where the third step substitutes the value of κ , p$_{α}$$_{+}$ = p − 1 for α > 0 and p$_{α}$$_{+}$ = p for α < 0. Case (ii) : When ( α < 0 , y ≤ µ ) or ( α > 0 , y > µ ), the cdf is, F ( y | θ ) =        y ∫ −∞ κ ( [ Φ ( v ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ] exp { − v $^{∗}$p$_{α}$$_{−}$ + 1 $_{2}$α$^{2}$p 2 α$_{−}$ } + Φ ( αp$_{α}$$_{+}$ − v ∗ α ) exp { − v $^{∗}$p$_{α}$$_{+}$ + 1 $_{2}$α$^{2}$p 2 α + } ) dv, if α < 0 , y ≤ µ 1 − ∞ ∫ y κ ( [ Φ ( v ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ] exp { − v $^{∗}$p$_{α}$$_{−}$ + 1 $_{2}$α$^{2}$p 2 α$_{−}$ } + Φ ( αp$_{α}$$_{+}$ − v ∗ α ) exp { − v $^{∗}$p$_{α}$$_{+}$ + 1 $_{2}$α$^{2}$p 2 α + } ) dv, if α > 0 , y > µ. (A.11) Note that in both the subcases of equation (A.11), the integral remains the same and only the limits of integration changes. Hence, we evaluate each terms individually over the limits ( a, b ) and will substitute values of ( a, b ) as per our requirement. To evaluate the first integral component of equation (A.11) denoted C$_{1}$ , we substitute z = v $^{∗}$/α − αp$_{α}$$_{−}$ and perform integration-by-parts as follows. C$_{1}$ = κ b ∫ a Φ ( v ∗ α − αp$_{α}$$_{−}$ ) exp { − v $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } dv = κ exp { − 1 2 α $^{2}$p2 α$_{−}$ } ∫ b ∗ α − αp$_{α}$$_{−}$ a ∗ α − αp$_{α}$$_{−}$ [ ασ Φ( z ) exp {− αp$_{α}$$_{−}$z } ] dz 31