  We study the Neyman-Pearson problem for convex expectations on
L^{\infty}(\mu). The existence of the optimal test is given. Without assuming
that the level sets of penalty functions are weakly compact, we prove that the
optimal tests for convex expectations on L^{\infty}(\mu) are just the classical
Neyman-Pearson tests between a fixed representative pair of simple hypotheses.
Then we show that the Neyman-Pearson problem for convex expectations on
L^{1}(\mu) can be solved similarly.
