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FRM Exam Part I · Sample Moments

Best Linear Unbiased Estimator (BLUE) Explained for FRM Part I

Updated 11 October 2026 · Fact-checked

A BLUE is the best linear unbiased estimator: linear in the data, unbiased (expected value equals the true parameter) and with the smallest variance among all such estimators. Under iid draws with finite variance, the sample mean is BLUE for the population mean, with variance σ² ÷ n.

Understand Best Linear Unbiased Estimator (BLUE)

An estimator is a rule that turns sample data into a guess for a population parameter. The sample mean is an estimator of the population mean. Because the sample is random, the estimator is itself a random variable with its own mean and variance. Exam questions ask about that distribution.

Unbiased means the expected value of the estimator equals the true parameter: E(θ̂) = θ. Bias = E(θ̂) − θ. An unbiased estimator is right on average, but any single estimate can still be far off.

Efficient means smallest variance among the unbiased estimators you are comparing. Lower variance means estimates cluster more tightly around the truth. Consistent means the estimator converges in probability to the true value as n grows. Its sampling distribution collapses onto the parameter. Unbiasedness is a fixed-sample property. Consistency is a large-sample property. They are different.

Linear means the estimator is a weighted sum of the observations: θ̂ = Σ wᵢXᵢ. The sample mean has all weights equal to 1/n. BLUE combines three properties: linear, unbiased and minimum variance within the class of linear unbiased estimators.

Why is the sample mean BLUE? Suppose the Xᵢ are iid with mean μ and variance σ². Unbiasedness forces the weights to sum to 1. Variance of Σ wᵢXᵢ is σ² Σ wᵢ². With weights summing to 1, Σ wᵢ² is smallest when all weights are equal at 1/n. So equal weighting wins. The same logic underlies the Gauss-Markov theorem for OLS: under its assumptions, OLS is BLUE.

Key formulas to remember

Bias
Bias(θ̂) = E(θ̂) − θ
Unbiased when bias equals zero.
Sample mean
X̄ = (1/n) Σ Xᵢ
A linear estimator with every weight equal to 1/n.
Expected value of sample mean
E(X̄) = μ
Holds for iid draws (really only needs each Xᵢ to have mean μ).
Variance and standard error of sample mean
Var(X̄) = σ² ÷ n; SE = σ ÷ √n
Needs iid (or uncorrelated) observations. Falls as n rises, which gives consistency.
Linear estimator conditions
θ̂ = Σ wᵢXᵢ; unbiased if Σ wᵢ = 1; Var = σ² Σ wᵢ²
Minimised at wᵢ = 1/n, so the sample mean is BLUE.
Mean squared error
MSE = Var(θ̂) + Bias²
A biased estimator can have lower MSE than an unbiased one.
Consistency
θ̂ → θ in probability as n → ∞
Sufficient: bias → 0 and variance → 0.

How to solve Best Linear Unbiased Estimator (BLUE) questions

Use this routine for any question on estimator properties or BLUE.

  1. 1Identify the estimator and write it as a formula. Check whether it is a weighted sum of the data (linear).
  2. 2Compute its expected value using E(Xᵢ) = μ. Compare with the true parameter to find the bias.
  3. 3Compute its variance using Var = σ² Σ wᵢ² for independent observations.
  4. 4If asked about efficiency, compare variances only among unbiased estimators.
  5. 5If asked about consistency, ask what happens to bias and variance as n → ∞. Both going to zero gives consistency.
  6. 6If asked about BLUE, check all three: linear, unbiased, smallest variance among linear unbiased estimators.
  7. 7State the conditions: iid or uncorrelated errors with constant variance, finite variance.
  8. 8Choose the option that matches your derived result.

Quickest way: Weights test

When to use it: When you must decide whether a linear estimator is unbiased or which of two is more efficient.

  1. Add the weights. If they sum to 1, the estimator is unbiased for μ. If not, bias = (Σ wᵢ − 1)μ.
  2. Square each weight and add. The smaller Σ wᵢ² has the lower variance.
  3. Equal weights 1/n give the minimum Σ wᵢ² = 1/n.
  4. For consistency, check whether Var → 0 as n grows, such as σ²/n.

Common mistakes in Best Linear Unbiased Estimator (BLUE)

  • Treating unbiased and consistent as the same thing.

    Both sound like 'correct on average or eventually'.

    Fix: Unbiased concerns the mean at a fixed n. Consistent concerns convergence as n → ∞. An estimator can be one without the other, such as X₁ alone (unbiased, not consistent).

  • Calling any unbiased estimator efficient.

    Students forget efficiency is about variance.

    Fix: Efficient means lowest variance among unbiased estimators. Compare variances before choosing.

  • Saying BLUE means smallest variance among all estimators.

    The word 'best' is read too broadly.

    Fix: BLUE is best only within linear unbiased estimators. Nonlinear or biased estimators may do better.

  • Forgetting the weights must sum to 1.

    Focus goes to the variance calculation.

    Fix: Always check Σ wᵢ = 1 first. If it fails, the estimator is biased.

  • Ignoring MSE and assuming bias is always bad.

    Unbiasedness is taught as the goal.

    Fix: Remember MSE = variance + bias². A small bias can be accepted for a large variance reduction.

  • Using σ² instead of σ²/n for the variance of the sample mean.

    Mixing up the variance of a single observation with that of the average.

    Fix: Divide variance by n, or the standard deviation by √n.

Worked examples

Example 1

X₁, X₂, X₃ are iid with mean μ and variance σ² = 36. Estimator A = (X₁ + X₂ + X₃)/3. Estimator B = 0.5X₁ + 0.3X₂ + 0.2X₃. Which is more efficient, and what is the variance of B?

Show the solution
  1. Weights of A sum to 1, weights of B sum to 0.5 + 0.3 + 0.2 = 1. Both are unbiased.
  2. Var(A) = σ² ÷ 3 = 36 ÷ 3 = 12.
  3. Σ wᵢ² for B = 0.25 + 0.09 + 0.04 = 0.38.
  4. Var(B) = 36 × 0.38 = 13.68.
  5. 12 < 13.68, so A has lower variance.

Answer: A (the sample mean) is more efficient. Var(B) = 13.68 versus Var(A) = 12.

Example 2

An estimator of μ is θ̂ = (X₁ + X₂ + … + Xₙ) ÷ (n + 1), with iid data of mean μ = 10. For n = 9, find the bias. Is the estimator consistent?

Show the solution
  1. E(θ̂) = nμ ÷ (n + 1).
  2. For n = 9: E(θ̂) = 9 × 10 ÷ 10 = 9.
  3. Bias = 9 − 10 = −1.
  4. General bias = nμ/(n+1) − μ = −μ/(n+1), which tends to 0 as n → ∞.
  5. Variance = nσ² ÷ (n+1)², which tends to 0 as n → ∞.
  6. Both bias and variance go to zero, so the estimator converges to μ.

Answer: The bias is −1 (it underestimates), but the estimator is consistent.

Exam tips

  • Read the wording closely: 'on average' points to unbiasedness, 'as the sample grows' points to consistency, 'smallest variance' points to efficiency.
  • When two estimators are offered, test unbiasedness through the weight sum before comparing variances.
  • Remember the sample mean is BLUE only under assumptions such as iid or uncorrelated draws with equal variance. Questions may test which assumption fails.
  • Link to OLS: the Gauss-Markov theorem says OLS is BLUE under the classical assumptions. Heteroskedasticity or serial correlation breaks the 'best' part, not unbiasedness.
  • Use MSE = variance + bias² when a question compares a biased and an unbiased estimator.

Practice questions from Sample Moments

Best Linear Unbiased Estimator (BLUE): frequently asked questions

Is the sample mean the best linear unbiased estimator?

Yes, for the population mean when observations are iid (or uncorrelated with equal variance and finite variance). Among all linear unbiased estimators, equal weights of 1/n give the smallest variance, σ²/n.

What is the difference between unbiasedness and consistency?

Unbiasedness means the estimator's expected value equals the true parameter at any sample size. Consistency means the estimator converges to the true value as the sample size grows. One does not imply the other.

What does efficient mean for an estimator?

An efficient estimator has the lowest variance among the unbiased estimators being compared. Lower variance gives tighter estimates around the true value.

Does BLUE appear in regression questions too?

Yes. The Gauss-Markov theorem states that under the classical assumptions, OLS estimators are BLUE. If errors are heteroskedastic or serially correlated, OLS stays unbiased but is no longer the best.