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
and
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
and
.