LRT(Likelihood Ratio Test)

LRT of size $\alpha$ reject $H_0$ if
$\frac{L(\hat{\theta}^{\Omega_0};\underline{x})}{L(\hat{\theta}^{MLE};\underline{x})} \leq k$
with $k \text{ satisfying } P_{\theta_0}(\frac{L(\hat{\theta}^{\Omega_0};\underline{x})}{L(\hat{\theta}^{MLE};\underline{x})} \leq k) = \alpha$

• log form
  LRT of size $\alpha$ reject $H_0$ if
  $2[l(\hat{\theta}^{\Omega_0};\underline{x})-l(\hat{\theta}^{\Omega};\underline{x})] \leq c$
  with $\underset{\theta \in \Omega_0}{max} P(l(\hat{\theta}^{\Omega_0};\underline{x})-l(\hat{\theta}^{\Omega};\underline{x}))=\alpha$

ex1) $X_1,...,X_n \overset{iid}{\sim} N(\mu,\sigma^2),$ $\sigma^2$ is known & $\mu \in \Omega=(-\infty,\infty)$
$H_0: \mu=\mu_0$ vs $H_1:\mu \neq \mu_0$

ex2) T-test
$X_1,...,X_n \overset{iid}{\sim} N(\theta_1,\theta_2),$ $\theta_2$ is unknown, $-\infty<\theta_1<\infty,$ $\theta_2>0$
$H_0: \theta_1=\theta_0$ vs $H_1:\theta_1 \neq \theta_0$

ex3) Chi-square test
$X_1,...,X_n \overset{iid}{\sim} N(\mu,\theta),$ $-\infty<\mu<\infty,$ $\theta>0$
$H_0: \theta=\theta_0$ vs $H_1:\theta \neq \theta_0$

ex4) F-test
$X_1,...,X_n \overset{iid}{\sim} N(\mu_1,\theta_1), \space Y_1,...,Y_m \overset{iid}{\sim} N(\mu_2,\theta_2)$
$H_0: \theta_1=\theta_2$ vs $H_1: \theta_1 \neq \theta_2$