Let and be independent unbiased estimators of with standard errors and (). The linear combination of and that is unbiased for with minimum variance is
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IIT JAM 2021 Question 14: Point and interval estimation
Answer and worked solution
Answer: (D)
For a weighted average of independent unbiased estimators, the variance is minimised by weighting each inversely to its own variance.
The optimal weight on is and on is . This is option (D). The weights use variances , not the standard errors .
The trap: (C)
Option (C) puts the larger weight on the noisier estimator; the efficient rule gives more weight to the estimator with the smaller variance, as in (D).
Plain-text version
Let α̂₁ and α̂₂ be independent unbiased estimators of α with standard errors σ₁ and σ₂ (σ₁ ≠ σ₂). The linear combination of α̂₁ and α̂₂ that is unbiased for α with minimum variance is (A) (σ₁/(σ₁+σ₂))α̂₁ + (σ₂/(σ₁+σ₂))α̂₂ (B) (σ₂/(σ₁+σ₂))α̂₁ + (σ₁/(σ₁+σ₂))α̂₂ (C) (σ₁²/(σ₁²+σ₂²))α̂₁ + (σ₂²/(σ₁²+σ₂²))α̂₂ (D) (σ₂²/(σ₁²+σ₂²))α̂₁ + (σ₁²/(σ₁²+σ₂²))α̂₂
Also written as: Let alpha_hat_1 and alpha_hat_2 be independent unbiased estimators of alpha with standard errors sigma_1 and sigma_2 (sigma_1 != sigma_2). The linear combination of alpha_hat_1 and alpha_hat_2 that is unbiased for alpha with minimum variance is
Point and interval estimation has come up 28 times in 6 exams since 2008.
What this question tests
This question is on Properties of estimators, part of Point and interval estimation in Statistics.
What makes an estimator good: unbiased, consistent and efficient.
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Question from the IIT JAM 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.