Unbiased estimator

An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter.

In other words, an estimator is unbiased if it produces parameter estimates that are on average correct.

Table of contents

Definition
Examples
Biased estimator
Bias
More details
Keep reading the glossary

Definition

Remember that in a parameter estimation problem:

we observe some data (a sample, denoted by $\xi$ ), which has been extracted from an unknown probability distribution;
we want to estimate a parameter $\theta _{0}$ (e.g., the mean or the variance) of the distribution that generated our sample;
we produce an estimate of $\theta _{0}$ (i.e., our best guess of $\theta _{0}$ ) by using the information provided by the sample $\xi$ .

The estimate $\widehat{\theta }$ is usually obtained by using a predefined rule (a function) that associates an estimate $\widehat{\theta }$ to each sample $\xi$ that could possibly be observed

The function is called an estimator.

Definition An estimator is said to be unbiased if and only ifwhere the expected value is calculated with respect to the probability distribution of the sample $\xi$ .

Examples

The following table contains examples of unbiased estimators (with links to lectures where unbiasedness is proved).

Estimator	Estimated parameter	Lecture where proof can be found
Sample mean	Expected value	Estimation of the mean
Adjusted sample variance	Variance	Estimation of the variance
OLS estimator	Linear regression coefficients	Gauss-Markov theorem
Adjusted sample variance of the OLS residuals	Variance of the error of a linear regression	Normal linear regression model

Biased estimator

An estimator which is not unbiased is said to be biased.

Bias

The bias of an estimator $\widehat{\theta }$ is the expected difference between $\widehat{\theta }$ and the true parameter:

Thus, an estimator is unbiased if its bias is equal to zero, and biased otherwise.

More details

Unbiasedness is discussed in more detail in the lecture entitled Point estimation.

Keep reading the glossary

Previous entry: Unadjusted sample variance

Next entry: Variance formula

How to cite

Please cite as:

Taboga, Marco (2021). "Unbiased estimator", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/unbiased-estimator.