GATE 2026 ST – Question 19
A sample X₁,X₂ has density f_α(x)=αx^(α−1)e^(−x^α),x>0, zero otherwise,α>0 unknown. Let X_(1)=min(X₁,X₂),X_(2)=max(X₁,X₂). Which statement is correct?
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Correct answer: (A) (X₁+X₂,X_(1)) is minimal sufficient.
Explanation
The official MCQ key is **A**, retained for grading. The likelihood α²(x₁x₂)^(α−1)exp(−x₁^α−x₂^α) has parameter-free ratios precisely for equal unordered samples; the order-statistic pair is minimal sufficient. Sum and minimum determine that pair, proving A. There is a source/key discrepancy: C is also true. Its signed difference d and minimum m reconstruct the ordered sample: if d≥0 then (x₁,x₂)=(m+d,m), otherwise (m,m−d). It is sufficient, but preserves the label/sign information discarded by the minimal statistic, so is not minimal. B can lose the smaller observation, and D is not complete since the sample-order sign has expectation zero for every α but is not almost surely zero.