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A UMP test maximizes power (Type II error) uniformly over all alternatives. By Neyman-Pearson, UMP tests exist for simple vs. simple hypotheses. For composite alternatives (e.g., H₁: θ > θ₀), UMP tests may not exist, but UMP unbiased tests sometimes do. The likelihood ratio test is often asymptotically UMP.
From the Neyman-Pearson Lemma, you know how to build the most powerful test when both the null and alternative are simple (single-point) hypotheses: reject H₀ when the likelihood ratio f(x; θ₁)/f(x; θ₀) exceeds a threshold c. This test is optimal in the sense that no other test at the same significance level α has higher power 1 − β, where β is the probability of missing a true alternative. The question that leads to UMP tests is: does this optimality extend when the alternative is composite — that is, when H₁ specifies a range of values like θ > θ₀?
A Uniformly Most Powerful (UMP) test is a test that, for every specific alternative value θ₁ in H₁, is the most powerful test at level α. "Uniformly" means the optimality holds simultaneously for the entire alternative region, not just at one point. For this to work, the critical region identified by Neyman-Pearson must be the same regardless of which specific θ₁ you plug in. When this happens, a single test achieves maximum power everywhere in H₁.
The condition that guarantees this is the monotone likelihood ratio (MLR) property. A family of distributions {f(x; θ)} has an MLR in a statistic T(x) if the likelihood ratio f(x; θ₁)/f(x; θ₀) is a non-decreasing function of T(x) whenever θ₁ > θ₀. In such families, the Neyman-Pearson critical region {T(x) > c} is the same for every θ₁ > θ₀, so the test is UMP for the one-sided alternative H₁: θ > θ₀. Exponential family distributions — Normal, Poisson, Binomial, Exponential — all have MLR in their natural sufficient statistic, which explains why clean one-sided tests exist for these distributions.
UMP tests generally do not exist for two-sided alternatives H₁: θ ≠ θ₀, because the most powerful test against θ₁ > θ₀ rejects in the right tail, while the most powerful test against θ₁ < θ₀ rejects in the left tail — no single critical region dominates both directions simultaneously. The resolution is UMP unbiased (UMPU) tests, which restrict attention to tests satisfying a bias condition (power ≥ α everywhere in H₁), and then find the most powerful unbiased test. For exponential families, UMPU tests for two-sided alternatives have critical regions in both tails, recovering the familiar two-sided t-test as a special case.