Demo: Detection Theory
For a Gaussian mean-shift test, decide $H_1$ when the sample mean exceeds a threshold. The plot connects the likelihood-ratio idea with false-alarm and detection probabilities.
Mathematical setup
Consider
\[H_0:X_i\sim N(0,\sigma^2), \qquad H_1:X_i\sim N(\mu_1,\sigma^2), \qquad \mu_1>0.\]For iid observations, the sample mean is sufficient for this one-sided mean-shift test:
\[\bar X\mid H_0\sim N\left(0,\frac{\sigma^2}{n}\right), \qquad \bar X\mid H_1\sim N\left(\mu_1,\frac{\sigma^2}{n}\right).\]The threshold rule is decide $H_1$ when $\bar X>\gamma$, so
\[P_{\mathrm{FA}}=\Pr_0(\bar X>\gamma), \qquad P_D=\Pr_1(\bar X>\gamma).\]Equivalently,
\[P_{\mathrm{FA}}=1-\Phi\left(\frac{\gamma\sqrt n}{\sigma}\right), \qquad P_D=1-\Phi\left(\frac{(\gamma-\mu_1)\sqrt n}{\sigma}\right).\]Sweeping $\gamma$ traces out the ROC curve.
What to try
- Move $\gamma$ right. False alarms decrease, but missed detections increase.
- Increase $\mu_1$ or $n$. The two statistic distributions separate more, improving the ROC.
- Increase $\sigma$. The distributions overlap more, so the same threshold becomes less decisive.
The operating probabilities are $P_{\mathrm{FA}}=P_0(\bar X>\gamma)$ and $P_D=P_1(\bar X>\gamma)$.