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GATE 2026 ST – Question 57

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 2 marks · Multiple select

From a single observation, test H₀:θ=1 against H₁:θ≠1 at level 0.05. For which density families does a UMP test exist?

  1. f_θ(x)=1/θ on 0<x<θ, zero otherwise; θ>0
  2. f_θ(x)=e^(−(x−θ)) on x>θ, zero otherwise; θ∈ℝ
  3. f_θ(x)=e^(−x/θ)/θ on x>0, zero otherwise; θ>0
  4. f_θ(x)=e^(−(x−1)/θ)/θ on x>1, zero otherwise; θ>0

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Correct answer: (A) f_θ(x)=1/θ on 0<x<θ, zero otherwise; θ>0; (B) f_θ(x)=e^(−(x−θ)) on x>θ, zero otherwise; θ∈ℝ

Explanation

For A, reject if x<0.05 or x>1. Under θ=1 its size is .05; for θ<1 the lower interval maximizes rejection, and for θ>1 every size-.05 part within (0,1) contributes equally while x>1 is free of null cost. For B, reject x<1 or x>1−ln(.05). The first part is null-impossible; on the common support a lower-location alternative has constant likelihood ratio, while upper alternatives favor the upper tail, giving one optimal test for both directions. C and D are scale-exponential families whose lower and upper alternatives demand opposite tails, so no common UMP test exists. **A,B**.

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