Generalized Likelihood Ratio Tests and Uniformly Most Powerful Tests

For testing a one-sided hypothesis in a one-parameter family of distributions, it is shown that the generalized likelihood ratio (GLR) test coincides with the uniformly most powerful (UMP) test, assuming certain monotonicity properties for the likelihood function.In particular, the equivalence of GLR tests and UMP tests holds for one-parameter exponential families.In addition, the relationship between GLR and UMPU (UMP unbiased) tests is considered when testing two-sided hypotheses.