Mathematical Foundations for Signal Processing, Communications, and Networking

11.6.2    UMP Detection with Both Composite Hypotheses

We now consider the more general case where both the hypotheses are composite. The UMP optimization problem can be stated as:

$\text{Maximize}{\text{P}}_{\text{D}}\left(\tilde{\delta };\theta \right),\text{for}\text{all}\theta \in {\text{Λ}}_1,\text{subject}\text{to}\underset{\theta \in {\text{Λ}}_0}{\text{sup}}{\text{P}}_{\text{F}}\left(\tilde{\delta };\theta \right)\le \alpha .$

(11.21)

If a UMP test δ̃_UMP exists, then it must satisfy the following conditions. First,

$\underset{{\theta }_0\in {\text{Λ}}_0}{\text{sup}}{\text{P}}_{\text{F}}\left({\tilde{\delta }}_{\text{UMP}};{\theta }_0\right)\le \alpha .$

(11.22)

Second, for any δ̃ ∈ Δ̃ that satisfies ${\sup }_{{\theta }_0\in {\text{Λ}}_0}{\mathrm{P}}_F\left(\tilde{\delta };{\theta }_0\right)\le \alpha$ we must have

${\text{P}}_{\text{D}}\left(\tilde{\delta };{\theta }_1\right)\le {\text{P}}_{\text{D}}\left({\tilde{\delta }}_{\text{UMP}};{\theta }_1\right)\text{for}\text{all}{\theta }_1\in {\text{Λ}}_1.$

(11.23)

The following example illustrates a case where a UMP solution can be found. Also see Exercises 11.9.12 and 11.9.13.

Example 11.6.4. Testing Between Two One-Sided Composite Signals in Gaussian Noise. This is an extension of Example 11.6.1 in which the observation ...