GATE 2023 ST – Question 63

Estimation · Unbiased and Minimum Variance Estimation · 2 marks · Numerical answer

The observations −1,−1/2,1,5/2,3 form a sample of size 5 from N(1/2,σ²),σ>0 unknown. T is an unbiased estimator of σ² attaining the Cramér–Rao lower bound. Find its observed value to two decimals.

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Correct answer: 2.60 to 2.90

Explanation

With the mean known,the efficient unbiased estimator is T=Σ(X_i−1/2)²/5; its variance 2σ⁴/5 equals the information bound. The squared deviations sum to 2.25+1+.25+4+6.25=13.75. Therefore T=13.75/5=**2.75**.

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