GATE 2025 ST – Question 62

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 2 marks · Numerical answer

X₁,X₂ are iid with density e^(x−θ) for x≤θ, zero otherwise,θ∈ℝ. For H₀:θ≥0 against H₁:θ<0, let β(θ) be the power of a level-.09 UMP test. Find β(ln .36) to two decimals.

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Correct answer: 0.67 to 0.72

Explanation

Let T=max X_i. Reject T<c, since smaller endpoints favor the alternative. For c≤θ, P_θ(T<c)=e^(2(c−θ)); otherwise it is 1. Under θ≥0 the maximum rejection occurs at θ=0. Size .09 gives e^(2c)=.09, so c=ln .3. At θ=ln .36>c, β=(.3/.36)²=25/36≈**.69**. The common lower-tail likelihood-ratio ordering makes this test UMP.

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