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?
Show 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.
If a uniformly most powerful test exists then its size is \(\alpha\)
If a uniformly most powerful unbiased test exists then it is necessarily a uniformly most powerful test
If a uniformly most powerful test exists then it is necessarily an unbiased test
Show Solution
The Correct Option isD
Solution and Explanation
Step 1: Set the definitions straight. A test of $H_0$ against $H_1$ has size $\alpha$ if it wrongly rejects a true $H_0$ with probability at most $\alpha$. Among all such tests, a uniformly most powerful, or UMP, test is one whose power curve sits above every other level $\alpha$ test's power curve, at every point of $H_1$.
Step 2: Test the claim that a UMP test always exists. This is too strong. UMP tests show up mainly in one sided problems with a monotone likelihood ratio. In many two sided problems, different level $\alpha$ tests win at different alternatives, so no single test dominates everywhere. Statement (A) fails.
Step 3: Test the claim about size equal to $\alpha$. Being UMP only means the power is maximised among level $\alpha$ tests, where level $\alpha$ allows the actual rejection probability under $H_0$ to be anywhere up to $\alpha$. Nothing in the definition forces this probability to land exactly on $\alpha$ in all cases, so the blanket statement in (B) fails.
Step 4: Test the claim linking UMP unbiased and UMP. The UMP unbiased test is chosen as the best performer only inside the smaller family of unbiased tests. When the full class of level $\alpha$ tests has no single winner, for instance a common two sided testing setup, restricting to unbiased tests can still produce a best one. That best unbiased test is not automatically best over the whole class, so (C) fails.
Step 5: Test the claim that a UMP test is unbiased. Take a UMP test $\phi$ of level $\alpha$. Set it against the plain test that ignores the data completely and rejects with fixed probability $\alpha$. This plain test still has level $\alpha$, and its power is flat at $\alpha$ everywhere on $H_1$. Because $\phi$ beats every level $\alpha$ test in power, $\phi$ must beat this plain one too, giving \[ \text{Power}_\phi(\theta) \geq \alpha \text{ for all } \theta \in H_1. \] Power never dropping below $\alpha$ is exactly what unbiasedness means. So (D) holds in general.
Step 6: Conclude. Only the statement tying UMP tests to unbiasedness survives the check.\[ \boxed{\text{If a UMP test exists then it is necessarily an unbiased test}} \]
