Introduction

In the realm of statistical estimation, the concept of “BLUE” holds significant importance. BLUE stands for Best Linear Unbiased Estimator, and it refers to a unique property that some estimators possess. In this chapter, we’ll delve into the meaning behind the statement that the mean estimator is BLUE. We’ll uncover why this property is prized in statistical analysis and how it influences the accuracy and efficiency of estimation.

BLUE Property

The term “BLUE” signifies an ideal scenario for an estimator in terms of both bias and efficiency. Let’s break down the components of the acronym:

• Best: The estimator is optimal in terms of its statistical properties, namely unbiasedness and efficiency.

• Linear: The estimator can be expressed as a linear combination of the observed data points.

• Unbiased: The estimator’s expected value equals the true parameter value it is estimating.

• Estimator: The estimator is a statistical tool used to estimate a population parameter based on sample data.

Significance of Unbiasedness

For an estimator to be BLUE, it must be unbiased. Unbiasedness means that, on average, the estimator’s estimates are centered around the true parameter value. This property is crucial as it ensures that the estimator is not systematically overestimating or underestimating the parameter.

Importance of Efficiency

In addition to being unbiased, a BLUE estimator is also efficient. Efficiency implies that among all unbiased estimators, the BLUE estimator has the smallest possible variance. In other words, it provides estimates that are the closest to the true parameter value on average.

Example: Consider a sample of stock returns and the need to estimate the population mean return. If an estimator meets the criteria of being both unbiased and efficient, it can be referred to as a BLUE estimator. The choice of the right estimator becomes paramount as it impacts the quality and reliability of investment decisions made based on the estimated mean return.