We will review the generalized Neyman-Pearson lemma, see [CK01].
Let be a probability space. are two subspaces of -integrable random variables. Let be defined by
The problem is to find
In other words, it is to maximize
over the constraints
The following theorem is a straightforward generalization of the result of [CK01].
Theorem 1 Suppose Assumption 1holds. Then, there exists , such that
- For all , following holds:
This implies the exitence of the saddle point, i.e.
- Let . Then,
Moreover, is given by
- For all , there exists such that
can be taken by and .
- can take the form of
where is chosen to satisfy
- In such a way, we have