GATE 2024 ST – Question 56

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

An iid exponential-scale sample n≥2 has density e^(−x/λ)/λ,x≥0,λ>0. Put T₁=ΣX_i,T₂=1/T₁. Write χ²_(ν,α) for the upper-tail-α quantile. Which test is UMP level α for H₀:λ=λ₀ against H₁:λ>λ₀?

Reject if (2/λ₀)T₁>χ²_(2n,α)
Reject if (2/λ₀)T₁>χ²_(2n,1−α)
Reject if (λ₀/2)T₂>χ²_(2n,α)
Reject if (λ₀/2)T₂>χ²_(2n,1−α)

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Correct answer: (A) Reject if (2/λ₀)T₁>χ²_(2n,α)

Explanation

For every λ>λ₀ the likelihood ratio increases with the sample sum, giving a common upper-tail most powerful region. Under the null, 2T₁/λ₀~χ²_(2n). Rejecting above its upper-tail-α quantile gives exact size α. **A**; the reciprocal statistic reverses the relevant ordering.

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