*16*

*Likelihood Ratio Tests*

*The critical values of the likelihood ratio test are usually based on an asymptotic approximation. We derive the asymptotic distribution of the likelihood ratio statistic and investigate its asymptotic quality through its asymptotic power function and its Bahadur efficiency*.

**16.1 Introduction**

Suppose that we observe a sample *X*_{1},..., *X*_{n} from a density *p*_{θ}, and wish to test the null hypothesis *H*_{0} : *θ* ∈ Θ_{0} versus the alternative *H*_{1} : *θ* ∈ Θ_{1}. If both the null and the alternative hypotheses consist of single points, then a most powerful test can be based on the log likelihood ratio, by the Neyman-Pearson theory. If the two points are *θ*_{0} and *θ*_{1}, respectively, then the optimal test statistic is given by

For certain ...