Generalized Likelihood Ratio Test

thm. Generalized Likelihood Ratio Testing (GLT). Comparing Hypotheses:

1. $H_0\in {\Theta }_0$

2. $H_1\in {\Theta }_1$ The Generalized Likelhood Ratio is:

$$\tilde{\Lambda }=\frac{{\sup }_{\theta \in {\Theta }_0\cup {\Theta }_1}\mathcal{L}(\theta )}{{\sup }_{\theta \in {\Theta }_0}\mathcal{L}(\theta )}\stackrel{\text{usually}}{=}\frac{\mathcal{L}({\theta }_0)}{\mathcal{L}({\hat{\theta }}_{\text{MLE}})}$$

The Generalized Likelihood Ratio Test (GLT) is:

$$\delta :\{\begin{array}{ll}{H}_0& \text{else }\\ H_1& \text{if }\tilde{\Lambda }>c\end{array}$$

\tilde \Lambda is often very hard to manipulate. Use Wilk’s phenomenon in order to approximate the cutoff regions.

• thm. Wilk’s Phenomenon. For Hypotheses

  1. $H_0\in {\Theta }_0$ ← $r_0$ dimentions

  2. $H_1\in {\Theta }_1$ ← $r_1$ dimensions i.e. $H_0\subset H_1$ then:

$$2\ln \Lambda \underset{d}{\stackrel{n\to \infty }{⟶}}{\chi }_{r_1-r_0}^2$$