FRM Part I · FRM Exam Part I · Sample Moments
Three i.i.d. observations X1, X2, X3 each have variance 12. Estimator A is the sample mean (X1+X2+X3)/3. Estimator B is (X1+2X2+3X3)/6. Both are unbiased for the population mean. What is the variance of B and how does it compare to A?
Estimator B's variance is 12 × 14/36 = 4.67, higher than the sample mean's 4. With equal weights the sum of squared weights is minimized subject to weights summing to one, so the sample mean is the more efficient linear unbiased estimator.
- AVariance 4.67, higher than A's variance of 4Correct
- BVariance 4.67, lower than A's variance of 4
- CVariance 14, higher than A's variance of 4
- DVariance 4, equal to A's variance
Explanation
Var(A) = 12/3 = 4. B's weights are 1/6, 2/6, 3/6, so the sum of squared weights is 14/36. Var(B) = 12 × 14/36 = 4.67, higher than 4. Option 14 forgets to divide by 36.
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