Introduction

In the realm of statistical estimation, understanding the bias of an estimator is paramount. Bias is a concept that sheds light on the accuracy and reliability of an estimator. In this chapter, we will delve into the intricacies of bias, exploring what it means, how it arises, and its significance in statistical analysis.

Bias of an Estimator

The bias of an estimator refers to the difference between the expected value of the estimator and the true value of the parameter being estimated. In simpler terms, bias indicates whether, on average, an estimator overestimates or underestimates the parameter it aims to estimate. A biased estimator tends to consistently deviate from the true value in a specific direction.

Mathematically, the bias (B) of an estimator can be expressed as:

$$B(\hat{\theta})=E(\hat{\theta})-\theta$$ Where:

• $\hat{\theta}$ is the estimator
• $E(\hat{\theta})$ is the expected value of the estimator
• $\theta$ is the true value of the parameter being estimated

Bias Measures

Bias is crucial because it provides insights into the systematic errors inherent in an estimator. If an estimator is unbiased, it means that, on average, it provides estimates that are close to the true parameter value. Conversely, a biased estimator consistently provides estimates that deviate from the true value by a certain amount.

Example: Consider a random sample of students’ heights. Let’s say we want to estimate the population mean height $(\mu)$ using the sample mean $(\bar{x})$ as an estimator. If the sample mean tends to consistently underestimate the true mean height, it indicates a negative bias. Conversely, if the sample mean consistently overestimates the true mean height, it indicates a positive bias.