Question

For testing a null hypothesis \(H_0\) against an alternative hypothesis \(H_1\) at level of significance \(\alpha \in (0,1)\), which of the following statements is correct?

• A. A uniformly most powerful test always exists
• B. If a uniformly most powerful test exists then its size is \(\alpha\)
• C. If a uniformly most powerful unbiased test exists then it is necessarily a uniformly most powerful test
• D. If a uniformly most powerful test exists then it is necessarily an unbiased test

Hint:

Compare a UMP test's power against the trivial test that rejects with fixed probability \(\alpha\); its power can never fall below \(\alpha\), which is the definition of unbiasedness.

Answer 1

The Correct Option is D

Step 1: Recall what level of significance and UMP test mean.
A test has level of significance \(\alpha\) if the chance of rejecting \(H_0\) when \(H_0\) is true is at most \(\alpha\). A test is called uniformly most powerful (UMP) at level \(\alpha\) if, among all tests of level \(\alpha\), it gives the highest power at every point of the alternative \(H_1\).

Step 2: Check option (A).
A UMP test does not always exist. It exists in special cases, such as one sided testing problems with a monotone likelihood ratio family, but for many two sided problems no single test beats every other level \(\alpha\) test at every alternative. So option (A) is FALSE.

Step 3: Check option (B).
The definition of a UMP test only asks that it is a level \(\alpha\) test with the largest power among level \(\alpha\) tests. Its actual size, the probability of rejecting \(H_0\) under \(H_0\), can be less than or equal to \(\alpha\); it is not forced to equal \(\alpha\) in every situation. So the blanket claim in (B) is FALSE.

Step 4: Check option (C).
A UMP unbiased test is the best test only among the smaller class of unbiased level \(\alpha\) tests. When no UMP test exists over the full class of all level \(\alpha\) tests, for example in many two sided problems, a UMP unbiased test can still exist within the unbiased class. So a UMP unbiased test need not be UMP over the entire class of tests. Option (C) is FALSE.

Step 5: Check option (D).
Suppose a UMP test \(\phi\) of level \(\alpha\) exists. Compare it with the trivial randomized test that rejects \(H_0\) with probability \(\alpha\) regardless of the data. This trivial test also has level \(\alpha\), and its power at any alternative in \(H_1\) is exactly \(\alpha\). Since \(\phi\) is UMP, its power at every alternative must be at least as large as the power of this trivial test, so
\[ \text{Power}_\phi(\theta) \geq \alpha \quad \text{for every } \theta \in H_1. \]
This is exactly the definition of an unbiased test: its power never falls below the significance level. So a UMP test is always unbiased, and option (D) is TRUE.

Final Answer:
A uniformly most powerful test, whenever it exists, always has power at least \(\alpha\) at every alternative, which makes it unbiased.\[ \boxed{\text{If a UMP test exists then it is necessarily an unbiased test}} \]