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GATE 2026 ST – Question 20

Estimation · Unbiased and Minimum Variance Estimation · 1 mark · Multiple choice

X₁,X₂ are independent N(θ,1),θ∈ℝ. Which listed statistic attains the Cramér–Rao lower bound for estimating its own expected value?

  1. X₁+X₂
  2. (X₁²+X₂²+5)/2
  3. (X₁+X₂)²
  4. 2X₁−X₂

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Show answer and explanation

Correct answer: (A) X₁+X₂

Explanation

Total Fisher information is 2. For A, expectation g=2θ, so bound=(g′)²/2=2 and variance=2, attaining it. B has variance 1+2θ² while its bound is 2θ²; C has variance 8+32θ² versus bound 32θ²; D has variance 5 versus bound 1/2. Only **A** attains its bound.

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