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.