BLUE (Best Linear Unbiased Estimator)

TL;DR

• An estimator that is a linear function of the observed data, has expected value equal to the true parameter, and is optimal among unbiased linear estimators in terms of variance.
• Common examples given are the sample mean (for the population mean) and the sample variance (for the population variance).
• Used widely across applied fields such as economics, finance, and engineering.

Definition

A best linear unbiased estimator (BLUE) is a statistical estimator that satisfies three conditions:

• It is a linear function of the data (expressible as a weighted sum of observed values with constant weights).
• It is unbiased (its expected value equals the true value of the parameter being estimated).
• It has the smallest variance among all unbiased linear estimators.

Explanation

• Linearity: The estimator must be a weighted sum of the observed values where the weights are constants. The sample mean is an example because it is the sum of observations divided by the sample size.
• Unbiasedness: The estimator’s expected value must equal the parameter being estimated. For the sample mean, the expectation equals the population mean, so it is unbiased.
• Minimum variance among unbiased linear estimators: Among all estimators that are both linear and unbiased, a BLUE attains the smallest variance, meaning it is the most efficient estimator within that class.

Examples

Sample mean as BLUE for the population mean

Suppose we have a sample of n observations from a normal distribution with mean μ and variance σ2. Using the sample mean x̄ to estimate μ:

$$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$$

Unbiasedness of the sample mean:

$$E(\bar{x}) = E\!\left(\frac{x_1 + x_2 + \dots + x_n}{n}\right) = \frac{E(x_1) + E(x_2) + \dots + E(x_n)}{n} = \mu$$

Variance of the sample mean:

$$\operatorname{Var}(\bar{x}) = \operatorname{Var}\!\left(\frac{x_1 + x_2 + \dots + x_n}{n}\right) = \frac{\operatorname{Var}(x_1) + \operatorname{Var}(x_2) + \dots + \operatorname{Var}(x_n)}{n^2}$$

Under independence and identical distribution, this gives:

$$\operatorname{Var}(\bar{x}) = \frac{\sigma^2}{n^2}$$

Because this variance is the smallest among unbiased linear estimators of the population mean (as presented), the sample mean is the BLUE of the population mean.

Other example mentioned

• The sample variance is stated as the BLUE of the population variance.

Use cases

• Estimators identified as BLUE are used to estimate population parameters in fields including economics, finance, and engineering.

Related terms

• Sample mean
• Sample variance
• Unbiased estimator
• Linear estimator
• Variance