The idea is that for small n the mean estimators for X/Y could be rather poor and one should restrain from using the information on Z to estimate Gamma.

Also I am asking for the epected value, that is imagine that after I set up this decision rule for the estimator many many coins are tossed and I have to minize the absolute loss.

This is very very important to me, if you have a hint for me please help me, it really is highly appreciated.

Yes the idea is that i have the information of n Gamma and Z tosses and now I am left with only a Z toss and try to find the better estimator for the Gamma coin which I know is dependent on the Z coin. The idea is that if I have a lot of samples with the same value for Z I probably should take the respective X/Y mean but when I have little experience for the result of Z I might be better of with the mean of Gamma.

It's the same Z as in the first and second line.

Thinking practically here, the (general) question is most interesting for relatively small n. How big does n need to be for the asymptotic $c/\sqrt(n)$ to hold?

This paper is just what I had been searching for. Thank you very much.