4.5.4 Likelihood Ratio Tests
We have already discussed the notion of likelihood L in Section 4.4. Likelihood ratio tests are a form of hypothesis tests that are popular due to theoretical results indicating they have strong power for a given significance level. They employ a ratio of likelihoods as their test statistic:
[4.42]
Here, the supremum is taken across all possible values for θ. The null hypothesis is generally rejected for small values of Λ.
It may be difficult to infer probabilities related to Λ, and hence to specify a non-rejection region for a desired significance level. A standard technique is to consider –2log(L). In many cases, this can be shown—see Lehmann and Romano (2005)—to be asymptotically centrally chi-squared with one degree of freedom as m→∞. That is, –2log(L) ~ χ2(1,0) for large m, assuming 
.