GATE 2025 ST – Question 21

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 1 mark · Multiple choice

For simple densities f₀,f₁, let φ be a most powerful level-.05 test based on X. Which is not necessarily correct?

φ is the unique most powerful level-.05 test.
E_f₁φ≥.05
E_f₀φ≤.05
For some c≥0, P_f(f₁(X)>cf₀(X))≤P_f(φ(X)=1) for both f

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Correct answer: (A) φ is the unique most powerful level-.05 test.

Explanation

Uniqueness can fail, especially where likelihood ratios tie or the supports are disjoint, so **A** is not guaranteed. The constant randomized test with rejection .05 is a competitor, proving B. C is the level condition. Neyman–Pearson optimality supplies a threshold c with certain rejection on the strict upper likelihood-ratio set, giving D; randomization may occur on the threshold boundary.

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