Efficiency

Definition

Efficient Estimator

$\mathrm{Var}(Y)=\frac{1}{nI(\theta )}$ where $I(\theta )$ is a Fisher Information

An unbiased estimator $Y$ is called an efficient estimator if its variance is the Rao-Cramer Lower Bound

Efficiency

Let $Y$ be an unbiased estimator, then the efficiency of $Y$, $0\le e(Y)\le 1$ is $e(Y)=\frac{1/nI(\theta )}{\mathrm{Var}(Y)}$

Asymptotic Efficiency

Let $X_1,X_2,\dots ,X_n$ be i.i.d. random variables with PDF $f(x|\theta )$, and ${\hat{\theta }}_1$ be an estimator satisfying $\sqrt{n}({\hat{\theta }}_1-{\theta }_0)\stackrel{D}{\to }N(0,{\sigma }_1^2)$, then

• the asymptotic efficiency of ${\hat{\theta }}_1$ is $e({\hat{\theta }}_1)=\frac{1/I({\theta }_0)}{{\sigma }_1^2}$
• If $e({\hat{\theta }}_1)=1$, then ${\hat{\theta }}_1$ is called asymptotically efficient
• Assume that another estimator${\hat{\theta }}_2$ such that $\sqrt{n}({\hat{\theta }}_2-{\theta }_0)\stackrel{D}{\to }N(0,{\sigma }_2^2)$ Then, the asymptotic relative efficiency (ARE) of ${\hat{\theta }}_1$ to ${\hat{\theta }}_2$ is $e({\hat{\theta }}_1,{\hat{\theta }}_2)=\frac{{\sigma }_2^2}{{\sigma }_1^2}$

Examples

In a Laplace distribution, ${\sigma }_m^2=1,{\sigma }_{\overline{X}}^2=2$ So, sample median is $2$ times more efficient than sample mean for the estimator of the location parameter.

In a Normal distribution, ${\sigma }_m^2=\frac{\pi }{2},{\sigma }_{\overline{X}}^2=1$ So, sample mean is $\frac{\pi }{2}$ times more efficient than sample median for the estimator of the location parameter.