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

It is a linear function of the data. This means that the estimator can be expressed as a weighted sum of the observed values, where the weights are constants. For example, the sample mean is a linear estimator because it can be written as the sum of the observed values divided by the sample size.

It is unbiased. This means that the estimator has an expected value that is equal to the true value of the parameter being estimated. For example, the sample mean is an unbiased estimator of the population mean because its expected value is equal to the true population mean.

It has the smallest variance among all unbiased linear estimators. This means that the estimator has the smallest possible variability around its expected value, which means that it is the most efficient estimator of the parameter. For example, the sample mean is the BLUE of the population mean because it has the smallest variance among all unbiased linear estimators of the population mean.

Examples of BLUE estimators are commonly used in statistical applications to estimate population parameters. For instance, the sample mean is the BLUE of the population mean, and the sample variance is the BLUE of the population variance. These estimators are widely used in many different fields, including economics, finance, and engineering.

Let’s consider an example to illustrate the concept of a BLUE estimator. Suppose we have a sample of n observations from a normal distribution with mean μ and variance σ2. We want to estimate the population mean μ using the sample mean x̄ as the estimator. The sample mean is a linear estimator because it can be written as the sum of the observed values divided by the sample size:

x̄ = (x1 + x2 + … + xn) / n

To check if the sample mean is unbiased, we need to calculate its expected value, which is the average value that it would take if we repeated the experiment many times. The expected value of the sample mean is equal to the true population mean because the sample mean is a weighted average of the observed values, where the weights are constants:

E(x̄) = E((x1 + x2 + … + xn) / n) = (E(x1) + E(x2) + … + E(xn)) / n = μ

Since the expected value of the sample mean is equal to the true population mean, the sample mean is an unbiased estimator of the population mean.

To check if the sample mean is the BLUE of the population mean, we need to compare its variance with the variance of other unbiased linear estimators of the population mean. The variance of the sample mean is given by the following formula:

Var(x̄) = Var((x1 + x2 + … + xn) / n) = (Var(x1) + Var(x2) + … + Var(xn)) / n2

Since the observations are independent and identically distributed, the variance of the sample mean is equal to the variance of the population divided by the sample size squared:

Var(x̄) = σ2 / n2

The variance of the sample mean is the smallest among all unbiased linear estimators of the population mean because it is inversely proportional to the sample size squared. This means that the sample mean is the most efficient estimator of the population mean because it has the smallest possible variability around its expected value. Therefore, the sample mean is the BLUE of the population mean.