FRM Part I · FRM Exam Part I · Sample Moments
An analyst estimates the population mean of i.i.d. returns using the sample mean. Which set of properties makes the sample mean the best linear unbiased estimator (BLUE) of the population mean?
The sample mean is BLUE because it is linear in the data, unbiased for the population mean, and has the smallest variance among all linear unbiased estimators. 'Best' is judged only within that linear, unbiased class, not against all possible estimators.
- AIt is a linear function of the observations, its expected value equals the population mean, and it has the smallest variance among all linear unbiased estimatorsCorrect
- BIt is a nonlinear function of the observations, is unbiased, and has the smallest mean squared error among all estimators
- CIt is a linear function of the observations, is biased downward in small samples, and has the smallest variance among all estimators
- DIt is a linear function of the observations and has the largest possible variance among unbiased estimators
Explanation
BLUE means best (minimum variance) within the class of linear and unbiased estimators. The sample mean is linear in the observations, has expected value equal to the population mean, and for i.i.d. data has the lowest variance among such estimators. The other options either drop unbiasedness, widen the class beyond linear estimators, or reverse the variance criterion.
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