GATE 2021 ST – Question 50

Estimation · Sufficiency and Completeness · 2 marks · Multiple choice

For an iid size-n sample from Uniform(0,θ),θ>0,n≥2,let X_(1),X_(n) be the extremes. P:(X_(1),X_(n)) is complete. Q:the range X_(n)−X_(1) is ancillary. Which hold?

P only
Q only
Both P and Q
Neither P nor Q

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Correct answer: (D) Neither P nor Q

Explanation

The nonzero function g=min−max/n has expectation θ/(n+1)−[nθ/(n+1)]/n=0 for every θ,so the pair is not complete. The range scales with θ and its mean is (n−1)θ/(n+1),so its law depends on θ and it is not ancillary. Neither holds. **D**.

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