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FRM Part I · FRM Exam Part I · Sample Moments

An analyst proposes estimating a population mean with the estimator 0.9 times the sample mean of i.i.d. observations, because it has a smaller variance than the sample mean. The true mean is 10 and the sample mean has variance 4. Which statement is correct?

Scaling by 0.9 gives an expected value of 9 against a true mean of 10, a bias of minus 1, and variance of 0.81 times 4, or 3.24. Because it is biased, it is not eligible under BLUE despite its lower variance.

  1. AThe estimator is unbiased with variance 3.24, so it beats the sample mean under BLUE
  2. BThe estimator has bias of -1 and variance 3.24, so it is not eligible under BLUECorrect
  3. CThe estimator has bias of +1 and variance 3.24, so it is not eligible under BLUE
  4. DThe estimator has bias of -1 and variance 4, so it is not eligible under BLUE

Explanation

Expected value is 0.9*10 = 9, so bias = 9 - 10 = -1. Variance = 0.81*4 = 3.24. Because it is biased, it is outside the class of unbiased linear estimators, so a lower variance does not make it a better BLUE candidate.

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