The GATE Grind

GATE 2023 ST – Question 24

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

For an iid size-n sample from Uniform(1/3,θ),θ>1/3,let Y=max X_i. Which estimator of θ is unbiased?

  1. ((n+1)/n)(Y−1/3)+1/3
  2. (n/(n+1))(Y−1/3)+1/3
  3. ((n+1)/n)(Y+1/3)−1/3
  4. Y

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Show answer and explanation

Correct answer: (A) ((n+1)/n)(Y−1/3)+1/3

Explanation

The scaled maximum (Y−1/3)/(θ−1/3) has Beta(n,1) distribution and mean n/(n+1). Solving its expectation relation for θ gives the bias correction in **A**.

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