Standard hypothesis testing for simple hypothesis with

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

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

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

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

\(p_{CR} +p_{FA} = 1\)

\(p_M + p_H = 1\)

\(R=\{x: \Lambda(x) < c\}\)

Loss and risk functions

0-1

The loss and risk functions are here with

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

The risk function can be plotted as

library(ggplot2)
ggplot(data.frame(x=1))+geom_blank()+labs(x='pFA',y='pH')

More general

The loss and risk functions are here

References

Knoblauch, K., & Maloney, L. T. (2012). Modeling Psychophysical Data in R. New York: Springer.

Maloney, L. T., & Zhang, H. (2010). Decision-theoretic models of visual perception and action. Vision Research, 50(23), 2362–2374.

Wickens, T. D. (2001). Elementary signal detection theory. Oxford University Press.