GATE 2026 ST – Question 43
An iid sample of size 3 has P_θ(X_i=1)=θ,P_θ(X_i=2)=1−θ,0<θ<1. Which statistic is not sufficient?
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Correct answer: (B) (X₁+X₂,X₁−X₃,X₂+X₃)
Explanation
The count of ones, equivalently total sum, is minimal sufficient. A and C reconstruct the entire sample; D supplies the total sum, so all are sufficient. B takes the same value (3,0,3) on samples (1,2,1) and (2,1,2), whose likelihood ratio is θ/(1−θ), depending on θ. Thus B is not sufficient. **B**.