For a bachelor thesis i'm trying to impose a new GARCH model using an non-linear exponentioal GARCH model with an underlying Generalized beta distribution of the second kind(introduced by McDonald(1984)). To implement this correctly I need the expectation of the absolute z, E|z|, where z is distributed as follow:
EGB2(z; delta, sigma, p, q) = ( Exp( p(z-delta)/sigma ) ) / ( |sigma|B(p,q)( 1+ exp( (z-delta)/sigma ) )^(p+q) )
Where B(p,q) is the beta function. (Because this is my first post i wasn't able to post a picture of the distribution. You can find the picture on page 523 of the article of Wang et al.)
I couldn't find anything on the internet, nor using calculus/probability theory books, so I was wondering if there is anyone who could help me out with this one.