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 \(σ^2\) .