content_image/322349.jpg content_image/322351.jpg content_image/322354.jpg
Case (i) : When ( α > 0 , y ≤ µ ) or ( α < 0 , y > µ ), the cdf is, 

<img src='content_image/322349.jpg'>

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, 

<img src='content_image/322351.jpg'>

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. 

<img src='content_image/322354.jpg'>

