This is the link to my pdf so you can read the problem :
The Gamma Problem -

It's a rought draft so please be kind to my bad english

The real problem arise in page 3 when im facing a 4x4 system i must solve,
the obstacle is Ek(log(zi+Y))

-if i try with monte carlo method to evaluate this integral, i generate Y and so sometimes, zi+Y happens to be negative because zi are negative and zi and Y are no longer related

-Also, if we simplify the problem and take gamma=alpha and delta=beta, and as an approximation of Ek(log(zi+Y)) we take Ek(log(Xi)) with current parameter estimation, we have that beta doesnt have an upper bound, so it's impossible to maximize,

-the only real solution would be to compute analytically Ek(log(zi+Y)), but i don't know how to do this...

thanks for helping