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 \((\Phi^-{1}(p_H), \Phi^{-1}(p_{FA}))\).