Likelihood Ratio Tests: Optimality and Extensions

This lecture covers Likelihood Ratio Tests (LRT) and their optimality in hypothesis testing. It discusses the breakdown of optimality theory in higher dimensions and general cases, leading to the combination of Neyman-Pearson paradigm with Maximum Likelihood. The lecture explores the definition of the Likelihood Ratio statistic, its application in simple vs simple settings, and the construction of reasonable tests. It also delves into the asymptotic distribution of the Likelihood Ratio, including Wilks' Theorem and its extensions to general cases. The content includes examples, proofs, and discussions on the significance level, asymptotic properties, and the relationship between Confidence Intervals and Hypothesis Tests.