When is the generalized likelihood ratio test optimal?

The generalized likelihood ratio test (GLRT), which is commonly used in composite hypothesis testing problems, is investigated. Conditions for asymptotic optimality of the GLRT in the Neyman-Pearson sense are studied and discussed. First, a general necessary and sufficient condition is established, and then based on this, a sufficient condition, which is easier to verify, is derived. A counterexample where the GLRT is not optimal, is provided as well. A conjecture is stated concerning the optimality of the GLRT for the class of finite-state sources.<
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