To construct critical regions for a test of composite hypothesis:

LRTs are generalizations of the Neyman-Pearson lemma that are not necessarily uniformly most powerful

LRTs compare the restricted maximum likelihood under $H_0$

against the unrestricted maximum likelihood over any value in the parameter space, $\theta \in \mathrm{\Theta }={\mathrm{\Theta }}_0\cup {\mathrm{\Theta }}_1$

Suppose we have a random sample $X_1,..,X_n\sim iidf(x;\theta )$. The maximum likelihood under $H_0$ and for all values of $\theta \in \mathrm{\Theta }$ are given by

where $\tilde{\theta }={\mathrm{\Theta }}_0$, the MLE of $\theta$ within ${\mathrm{\Theta }}_0$ (restricted MLE) and $\hat{\theta }$ is the unrestricted MLE of $\theta$

The likelihood ratio statistic 109,110

If $\mathrm{\Lambda }\approx 0$ we would like to reject $H_0$
If $\mathrm{\Lambda }\approx 1$ we would like to accept $H_0$

If $\mathrm{\Theta }={\mathrm{\Theta }}_0\cup {\mathrm{\Theta }}_1$, ${\mathrm{\Theta }}_0\cap {\mathrm{\Theta }}_1=\mathrm{\varnothing }$ and

then the critical region $\mathrm{\Lambda }\le k$, $0<k<1$ is a likelihood ratio test for testing $H_0$ against $H_a$

If we don't know the distribution of $\mathrm{\Lambda }$, but have a large sample size $n$:

If the samples come from a normal distribution, $-2[l(\tilde{\mu })-l(\hat{\mu })]\sim {\chi }_1^2$ exactly

• or just $2[l(\hat{\mu }-l(\tilde{\mu }))]$

Examples

111,112

Removed bc from assignment
:(