Unbiased estimator

Summary

• Given:
• A sample $x_1,x_2,...,x_n$
• A statistical model where $θ$ is an unknown parameter
• An estimator $\hat{θ}=g(x_1,…,x_n)$
• We say that $\hat{θ}$ is an unbiased estimator of $θ$ if $E\left( \hat{θ} \right)=θ$ .

Example

• $x_1,x_2,...,x_n$ is an IID random sample where each $x_i \sim N\left( μ,σ^2 \right)$ .
• Then $\bar{x}$ is an unbiased estimator of $μ$ and $s^2$ is an unbiased estimator of $σ^2$ .