For a one-sided alternative H₁: θ > θ₀, a test that rejects when T(x) > c is guaranteed to be UMP when:
AT(x) is any sufficient statistic for θ
BThe likelihood ratio f(x; θ₁)/f(x; θ₀) is a non-decreasing function of T(x) for all θ₁ > θ₀ — the monotone likelihood ratio (MLR) property
CThe p-value is minimized by this critical region across all alternatives
DThe test has the smallest Type I error rate among all tests of level α
The MLR condition ensures that the Neyman-Pearson critical region {T(x) > c} is the same regardless of which specific θ₁ > θ₀ you substitute into the likelihood ratio. When the critical region doesn't change with θ₁, a single test achieves maximum power at every point in H₁ — making it uniformly most powerful. Sufficiency alone does not guarantee this; the ordering of the likelihood ratio must be monotone in T(x).
Question 2 Multiple Choice
A student claims: 'For testing H₀: μ = 0 vs. H₁: μ ≠ 0 in a normal model, the two-sided t-test is UMP because it is optimal in both directions.' What is wrong with this claim?
AThe student is correct; the two-sided t-test is the UMP test for all normal testing problems
BNo UMP test exists for two-sided alternatives: the most powerful test against μ > 0 rejects in the right tail, while the most powerful test against μ < 0 rejects in the left tail — these are incompatible critical regions, so no single test dominates both directions
CThe t-test fails because it uses the wrong test statistic for this problem
DUMP tests only exist for non-normal distributions
This is the central limitation of UMP theory. 'Uniformly most powerful' requires a single critical region that achieves maximum power for every θ₁ in H₁. For two-sided H₁, the optimal critical regions for θ₁ > θ₀ and θ₁ < θ₀ point in opposite directions — a right-tail rejection region and a left-tail rejection region cannot both be optimal simultaneously. The two-sided t-test is the UMPU (unbiased) test, not the UMP test, because UMP doesn't exist.
Question 3 True / False
A UMP test for H₁: θ > θ₀ achieves maximum power simultaneously at every specific value θ₁ > θ₀, not just at a single chosen alternative.
TTrue
FFalse
Answer: True
The word 'uniformly' in UMP means exactly this: the optimality holds simultaneously for all alternatives in H₁, not just pointwise at one value. This is stronger than simply being the most powerful test at a single θ₁. The MLR property is what makes uniform optimality achievable — because the critical region {T(x) > c} doesn't change as θ₁ varies, it is simultaneously optimal everywhere in the one-sided alternative region.
Question 4 True / False
The likelihood ratio test is typically a UMP test, regardless of the form of the null and alternative hypotheses.
TTrue
FFalse
Answer: False
The likelihood ratio test is a broadly useful procedure and is often asymptotically optimal, but it is not always UMP. For two-sided alternatives, no UMP test exists at all. For one-sided alternatives in exponential families, the likelihood ratio test often coincides with the UMP test — but this is because of the MLR property in those families, not because likelihood ratio tests are UMP by definition. The claim overstates the generality of UMP optimality.
Question 5 Short Answer
Why do UMP tests generally fail to exist for two-sided alternatives, and what is the standard resolution?
Think about your answer, then reveal below.
Model answer: For a two-sided alternative H₁: θ ≠ θ₀, the most powerful test against θ₁ > θ₀ requires a right-tail critical region, while the most powerful test against θ₁ < θ₀ requires a left-tail critical region. These are mutually exclusive, so no single test can be most powerful in both directions simultaneously — UMP doesn't exist. The resolution is the UMP unbiased (UMPU) test, which restricts to tests with power ≥ α everywhere in H₁ and then finds the most powerful unbiased test, typically yielding a two-tailed critical region (recovering the familiar two-sided t-test for normal data).
The non-existence of UMP for two-sided tests is not a failure of the theory — it is a correct description of a genuine impossibility. The UMPU framework recovers tractable optimality by relaxing 'most powerful among all tests' to 'most powerful among unbiased tests.' This is a useful tradeoff: UMPU tests exist for all exponential family problems and include the standard two-sided tests as special cases.