Formula sheet

For $\theta \in \Theta_0$ it is the Type I error rate; for $\theta \in \Theta_1$ it is one minus the Type II error rate.

For a test that rejects for large $T$. Reject at level $\alpha$ exactly when $p \le \alpha$. For a simple null and continuous $T$, $p(\mathbf{X}) \sim \mathrm{Unif}(0,1)$ under $H_0$. It is not the probability that $H_0$ is true.

For simple $H_0\colon \theta = \theta_0$ versus simple $H_1\colon \theta = \theta_1$, with $k \ge 0$ and $\gamma$ chosen so that $E_{\theta_0}[\phi] = \alpha$, this likelihood-ratio test is most powerful at level $\alpha$.

With MLR in $T$, this is UMP of level $\alpha$ for $H_0\colon \theta \le \theta_0$ versus $H_1\colon \theta > \theta_0$. For discrete $T$, randomize at $T = c$ to get size exactly $\alpha$. Two-sided alternatives usually have no UMP test.

$0 \le \Lambda \le 1$. The denominator uses the unrestricted MLE and the numerator the MLE under $H_0$.

If, for each $\theta_0$, the acceptance region $A(\theta_0)$ gives a level-$\alpha$ test of $H_0\colon \theta = \theta_0$, then $P_\theta\big(\theta \in C(\mathbf{X})\big) \ge 1 - \alpha$ for every $\theta$: $C$ is a $1-\alpha$ confidence set. Conversely, a $1-\alpha$ confidence set $C$ gives the level-$\alpha$ acceptance regions $A(\theta_0) = \{\mathbf{x} : \theta_0 \in C(\mathbf{x})\}$.