Likelihood Ratio Tests

The subject of this chapter is the testing of hypotheses within regular parametric models. Thereby, we restrict ourselves to generalized likelihood ratio tests. As with the method of maximum likelihood, these tests presuppose densities with respect to some sigma-finite dominating measure. The chapter starts with compiling basic notions, such as hypothesis, alternative, statistical test, test statistic, critical value, type I and type II error, power function, asymptotic level, and consistency. These concepts are illustrated with the one-sided binomial test. Then, the Neyman-Person likelihood ratio, which leads to optimal tests if both the hypothesis and the alternative are simple, serves as a motivation to consider the generalized likelihood ratio (GLR) for testing a hypothesis within a general parametric model. The main result of this chapter is the asymptotic chi-square distribution of the logarithmically transformed GLR under the hypothesis. Further topics are the consistency of the GLR test, the close relation with chi-square tests, tests of independence in contingency tables, and the parametric bootstrap.