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Interpret the claim that the estimator of the mean is BLUE.

FRM® Part I · Quantitative Analysis · Sample Moments · FRM1.QA.5.e

The idea

Saying the sample mean is BLUE means it is the Best Linear Unbiased Estimator of the population mean. Linear means a weighted sum of the observations. Unbiased means the weights sum to one, so the expected value equals the mean. Best means smallest variance. For uncorrelated observations with a common mean and a common variance, the variance of a weighted sum is: Variance = σ² × Σw². This is minimised by equal weights of 1 ÷ n, giving the sample mean. No normality is required. The claim is limited. It says nothing about nonlinear estimators such as the median, which can be more precise with heavy tails, and it fails if variances differ or observations are correlated.

Formulas

Variance of a weighted average of uncorrelated observations

Var = σ² × Σ wᵢ²

σ
the common standard deviation of each observation
wᵢ
the weight on observation i, with the weights summing to one

Requires uncorrelated observations with a common mean and variance. Equal weights of 1 ÷ n minimise it, which is why the sample mean is BLUE.

A practice question

From the question bank, with the reason each wrong option is wrong.

The sample mean is described as the BLUE estimator of the population mean. Which interpretation is correct?

  1. Among all linear unbiased estimators of the mean, it has the smallest variance. - correct
  2. Among all estimators, linear or not, it has the smallest mean squared error.Why not: BLUE restricts the competition to linear unbiased estimators; nonlinear or biased estimators are not covered.
  3. It is the unbiased estimator with the lowest bias among linear estimators.Why not: Every estimator in the class is already unbiased; 'best' refers to minimum variance, not bias.
  4. It is unbiased and normally distributed in every sample size.Why not: BLUE makes no claim about the distribution of the estimator; normality is a large-sample CLT result.

BLUE = Best Linear Unbiased Estimator. 'Linear' means a weighted sum of the observations, 'unbiased' means the expected value is the parameter, and 'best' means the smallest variance within that class.

A harder one

Three independent observations share a mean and a variance sigma². Estimator A is the equal-weighted mean. Estimator B uses weights 0.5, 0.3 and 0.2. Compared with A, the variance of B is approximately how many times as large?

  1. 0.88Why not: Inverts the ratio, expressing A's variance relative to B's.
  2. 1.00Why not: Reasons that both are unbiased and so have the same variance; unbiased estimators can differ widely in variance.
  3. 1.14 - correct
  4. 1.50Why not: Compares the largest weight with the equal weight instead of the sums of squared weights.

Variance of a weighted sum = sigma² x (sum of squared weights). B: 0.25 + 0.09 + 0.04 = 0.38. A: 3 x (1/3)² = 0.3333. Ratio = 1.14, so equal weights is better, as BLUE says.

The hardest tier

Two uncorrelated observations have the same mean but variances of 1 and 4. Their equal-weighted average has variance 1.25. Which statement is correct?

  1. The equal-weighted mean is not BLUE here; weights of 0.8 on the low-variance observation and 0.2 on the other give an unbiased estimator with variance 0.80. - correct
  2. The equal-weighted mean is still BLUE, because it is unbiased and linear, with variance 1.25.Why not: Ignores that BLUE also requires equal variances; unequal variances allow a lower-variance linear unbiased combination.
  3. The equal-weighted mean is not BLUE; weights of 0.2 on the low-variance observation and 0.8 on the other give variance 2.60.Why not: Puts the larger weight on the noisier observation (and the variance computed is higher than for equal weights).
  4. The equal-weighted mean is not BLUE; weights of 0.8 and 0.2 give variance 1.60.Why not: Computes the variance with weights rather than squared weights, giving a figure worse than equal weights.

Minimum variance weights are inversely proportional to variance: 0.8 and 0.2. Variance = 0.8² x 1 + 0.2² x 4 = 0.80, below 1.25. BLUE for the plain average needs equal variances.

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