Standard hypothesis testing for simple hypothesis with

$$P_{\theta\ \in \Theta_N}(d(X)=a_N) = P(a_N | H_N)= P(Correct \ Rejection) = p_{CR}$$

$$P_{\theta\ \in \Theta_N}(d(X)=a_S) = P(a_S | H_N)= P(False \ Alarm) = p_{FA}$$

$$P_{\theta\ \in \Theta_S}(d(X)=a_N) = P(a_N | H_S)= P(Miss) = p_{M}$$

$$P_{\theta\ \in \Theta_S}(d(X)=a_S) = P(a_S | H_S)= P(Hit) = p_{H}$$

$$p_{CR} +p_{FA} = 1$$

$$p_M + p_H = 1$$

$$R=\{x: \Lambda(x) > \beta\}$$

Loss and risk functions

0-1

The loss and risk functions are here with

$$R(\theta,a)= \left\{ \begin{array}{ll} \beta(\theta) = P_{\theta\ \in \Theta_n}\left(d(X)=a_s\right) = P(a_s|H_0)=P(False \ Alarm) =p_{FA} = 1 - p_{CR}\\ 1 - \beta(\theta) = P_{\theta\ \in \Theta_s}\left(d(X)=a_n\right) = P(a_n|H_s)=P(Miss)=p_M = 1- p_H\\ \end{array} \right.$$

More general

The loss and risk functions are here

ROC curves

The performance of the observer could be summarized as a point in the space $$(p_H, p_{FA})$$. If the obsever performs the task under different conditions, this will produce different points in this space.

zROC curves

The space considered is $$(\Phi^-{1}(p_H), \Phi^{-1}(p_{FA}))$$.