FRM Part I · FRM Exam Part I · Sample Moments
An estimator of a population mean is defined as 0.25X1 + 0.25X2 + 0.25X3 + 0.25X4 + 0.10 for i.i.d. observations with mean 8 and variance 16. Which statement is correct?
The estimator is biased by 0.10 because the added constant shifts its expected value to 8.10. Its variance is still 4, since constants add no variance, but BLUE requires unbiasedness, so it fails.
- AIt is biased with bias 0.10 and cannot be BLUE, although its variance is 4Correct
- BIt is unbiased with variance 4 and is BLUE
- CIt is biased with bias 0.10 and variance 4.1
- DIt is unbiased because weights sum to one, with variance 1
Explanation
Expected value = 8 × (4 × 0.25) + 0.10 = 8.10, so bias is 0.10. The constant does not change variance: 16 × 4 × 0.0625 = 4. Because it is biased, it is outside the class of unbiased estimators and cannot be BLUE.
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