  We study the Neyman-Pearson theory for convex expectations (convex risk
measures) on $L^{\infty}(\mu)$. Without assuming that the level sets of penalty
functions are weakly compact, a new approach different from the convex duality
method is proposed to find a representative pair $(Q^{\ast },P^{\ast})$ such
that the optimal tests are just the classical Neyman-Pearson tests between the
representative probabilities $Q^{\ast}$ and $P^{\ast}$. The key observation is
that the feasible test set is compact in the weak$^{\ast}$ topology by a
generalized result of Banach-Alaoglu theorem. Then the minimax theorem can be
applied and the representative probability $Q^{\ast}$ is found first. Secondly,
under the probability $Q^{\ast}$, we find the representative probability
measure $P^{\ast}$ by solving a dual problem. Finally, we apply our results to
a shortfall risk minimizing problem in an incomplete financial market.
