content_image/306983.jpg content_image/306985.jpg content_image/306986.jpg content_image/306987.jpg content_image/306988.jpg
Secondly, we integrate the expression for M$_{2}$ as follows, 

<img src='content_image/306983.jpg'>

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, 

<img src='content_image/306985.jpg'>

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 . 

<img src='content_image/306986.jpg'>

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, 

<img src='content_image/306987.jpg'>

Case (ii) : When α < 0 and y ∗ ≤ 0 (i.e., y ≤ µ ), we have the following three components of the mgf , 

<img src='content_image/306988.jpg'>

