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GATE 2022 ST – Question 19

Estimation · Sufficiency and Completeness · 1 mark · Multiple choice

For an iid sample of size n from f(x;θ),which statement is NOT true?

  1. ΣX_i is complete sufficient for Poisson(θ),θ>0.
  2. (ΣX_i,ΣX_i²) is complete sufficient for N(θ,θ²),θ>0.
  3. The density θx^(θ−1),0<x<1,has monotone likelihood ratio in ∏X_i.
  4. The range X_(n)−X_(1) is ancillary for Uniform(θ,θ+1),θ>0.

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Correct answer: (B) (ΣX_i,ΣX_i²) is complete sufficient for N(θ,θ²),θ>0.

Explanation

A is a full one-parameter exponential family. In C,the likelihood ratio for θ₂>θ₁ is a positive constant times (∏X_i)^(θ₂−θ₁),hence increasing. Translation leaves the range unchanged,proving D. B is sufficient but not complete:write T₁=ΣX_i,T₂=ΣX_i². For n>1,E T₁²=n(n+1)θ² and E T₂=2nθ²,so T₁²−(n+1)T₂/2 is a nonzero statistic of mean zero for every θ. Thus **B**.

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