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## 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 X1,..., Xn from a density pθ, and wish to test the null hypothesis H0 : θ ∈ Θ0 versus the alternative H1 : θ ∈ Θ1. 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 θ0 and θ1, respectively, then the optimal test statistic is given by

For certain ...

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