Demo: Detection Theory Examples
Switch between two common textbook calculations: a signed mean-shift statistic and an energy statistic for a variance change. The point is to notice that different alternatives call for different sufficient statistics.
Mathematical setup
Mean-shift example:
\[H_0:X_i\sim N(0,1), \qquad H_1:X_i\sim N(\mu_1,1), \qquad T=\bar X.\]For the threshold rule $T>\gamma$,
\[P_{\mathrm{FA}}=1-\Phi(\gamma\sqrt n), \qquad P_D=1-\Phi((\gamma-\mu_1)\sqrt n).\]Variance-change example:
\[H_0:X_i\sim N(0,1), \qquad H_1:X_i\sim N(0,\sigma_1^2), \qquad T=\sum_{i=1}^n X_i^2.\]Under $H_0$, $T\sim\chi_n^2$; under $H_1$, $T/\sigma_1^2\sim\chi_n^2$, or equivalently $T\sim\sigma_1^2\chi_n^2$. For the rule $T>\gamma$,
\[P_{\mathrm{FA}}=\Pr(\chi_n^2>\gamma), \qquad P_D=\Pr\left(\chi_n^2>\frac{\gamma}{\sigma_1^2}\right).\]The displayed variance-change probabilities use these chi-squared survival functions.
What to try
- In mean-shift mode, raise $\mu_1$ and see detection improve without changing the false-alarm calculation for a fixed threshold.
- In variance-change mode, raise $\sigma_1$. The energy distribution under $H_1$ shifts right, so large-energy tests become more powerful.
- Compare threshold scales across the two modes. A reasonable threshold for $\bar X$ is not a reasonable threshold for $\sum_i X_i^2$.
For the variance-change example, the energy statistic uses the scaled chi-squared distribution: under $H_0$, $\sum_i X_i^2\sim\chi_n^2$, and under the larger-variance alternative, $\sum_i X_i^2/\sigma_1^2\sim\chi_n^2$.