So for my Neyman-Pearson test, I have already calculated the sufficient statistic $T$. It is distributed as following:
$$ T_{|H_{0}} = \text{Rayleigh}(\sigma)$$ $$ T_{|H_{1}} = \text{Rice}(A, \sigma)$$
It is clear how to design the detector to detect $H_{1}$. In other words, we want to find the threshold $\gamma$ such that we minimize the probability of false alarm:
$$ P_{FA|H_{1}} = Pr\left( T \geq \gamma | H_{0} \right)$$
This means our detection probability is:
$$ P_{D|H_{1}} = Pr\left( T \geq \gamma | H_{1} \right)$$
However, I want to find $\gamma$ such that I detect $H_{0}$ with a minimal $P_{FA}$. Would it just be:
$$ P_{FA|H_{0}} = Pr\left( T \geq \gamma | H_{1} \right)$$ $$ P_{D|H_{0}} = Pr\left( T \geq \gamma | H_{0} \right)$$
In other words, would I just switch the hypotheses? Or would it be:
$$ P_{FA|H_{0}} = Pr\left( T \leq \gamma | H_{1} \right)$$ $$ P_{D|H_{0}} = Pr\left( T \leq \gamma | H_{0} \right)$$
Edit:
I am using $P_{FA|H_{j}}$ to denote the probability of false alarm, given that I want to detect hypothesis $H_{j}$. Similary, $P_{D}$ stands for the probability of detection.
$Pr\left( T \leq \gamma | H_{j} \right)$ means simply the probability that the test statistic $T$ is less than or equal to $\gamma$ given $H_{j}$ is true.
I want to know what the forms of $P_{FA|H_{0}}$ and $P_{D|H_{0}}$ should be. Ie how should I determine the false alarm and detection probability given that I want to detect $H_{0}$? Also, can I choose $\gamma$ to minimize the probability of false alarm given that I want to detect $H_{0}$?
The hypothesis test I was looking for was to detect a true negative while minimizing the probability of false negatives.
In other words, I want to compute:
$$ P_{D|H_{0}} = P_{TN} = P\left( T \leq \gamma | H_{0} \right) $$
where I choose gamma to minimize:
$$ P_{FN} = P\left( T \leq \gamma | H_{1} \right) $$
This is essentially the reverse hypothesis test, which is what I was going for. This allows me to optimally detect when the null hypothesis is true.