Efficiency

Summary

• Given:
• A sample \(x_1,x_2,...,x_n\)
• A statistical model where \(θ\) is an unknown parameter
• Two different unbiased estimators of \(θ\) : \({\hat{θ}}_1\) and \({\hat{θ}}_2\)
• We say that \({\hat{θ}}_1\) is more efficient than \({\hat{θ}}_2\) if \(Var\left( {\hat{θ}}_1 \right)≤Var\left( {\hat{θ}}_2 \right)\)

Example

• \(x_1,x_2,...,x_n\) is an IID random sample where each \(x_i \sim N\left( μ,σ^2 \right)\) ( \(n\) is even). Then \(\bar{x}\) as well as

\[\bar{x}_{n/2}= \frac{1}{n/2}\sum_{i=1}^{n/2}{ ‍ }x_i\]

• are unbiased. However,

\[Var\left( \bar{x}_{n/2} \right)= \frac{σ^2}{n/2}> \frac{σ^2}{n}=Var\left( \bar{x} \right)\]

• so \(\bar{x}\) is more efficient.