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FRM Part I · FRM Exam Part I · Sample Moments

An analyst estimates a population mean using the sample mean of n independent, identically distributed observations with finite variance. Which statement correctly describes why the sample mean is called the best linear unbiased estimator (BLUE) of the population mean?

The sample mean is BLUE because, among all estimators that are linear combinations of the observations and unbiased for the population mean, it has the smallest variance. The restriction to linear unbiased estimators matters; it is not claimed to beat every biased or nonlinear estimator.

  1. AAmong all estimators that are linear in the observations and unbiased, it has the smallest varianceCorrect
  2. BAmong all unbiased estimators, linear or not, it has the smallest bias in small samples
  3. CAmong all linear estimators, it has zero variance as n grows large
  4. DAmong all estimators, it always has the lowest mean squared error, even if biased

Explanation

BLUE means minimum variance within the class of linear unbiased estimators. The sample mean is linear in the data, has expected value equal to the population mean, and has variance sigma^2/n, the lowest in that class. The claim about all estimators is wrong because biased estimators can have lower MSE, and 'smallest bias' is meaningless since unbiased estimators all have zero bias.

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