Efficiency (statistics)

In statistics, efficiency is one measure of desirability of an estimator. The efficiency of an unbiased statistic ${\displaystyle T}$ is defined as

${\displaystyle e(T)={\frac {1/{\mathcal {I}}(\theta )}{\mathrm {var} (T)}}}$

where ${\displaystyle {\mathcal {I}}(\theta )}$ is the Fisher information of the sample. Thus ${\displaystyle e(T)}$ is the minimum possible variance for an unbiased estimator divided by its actual variance. The Cramér-Rao bound can be used to prove that ${\displaystyle e(T)\leq 1}$:

${\displaystyle \mathrm {var} \left({\widehat {\theta }}\right)\geq {\frac {1}{{\mathcal {I}}(\theta )}}}$
${\displaystyle 1\geq {\frac {1/{\mathcal {I}}(\theta )}{\mathrm {var} \left({\widehat {\theta }}\right)}}\to 1\geq e(T)}$

Efficient estimator

If an unbiased estimator of a parameter ${\displaystyle \theta \in \Theta }$ attains ${\displaystyle e(T)=1}$ for all values of the parameter, then the estimator is called efficient.

Equivalently, the estimator achieves equality on the Cramér-Rao inequality for all ${\displaystyle \theta \in \Theta }$.

An efficient estimator is also a minimum variance unbiased estimator. This is because an efficient estimator maintains equality on the Cramér-Rao inequality for all parameter values, which means it attains the minimum variance for all parameters (the definition of an MVU estimator).

An MVU estimator is not necessarily efficient, because "minimum" does not mean equality holds on the Cramér-Rao inequality.

Asymptotic efficiency

For some estimators, they can attain efficiency asymptotically and are thus called asymptotically efficient estimators. This can be the case for some maximum likelihood estimators or for any estimators that attain equality of the Cramér-Rao bound asymptotically.

Examples

Consider a sample of size ${\displaystyle N}$ drawn from a normal distribution of mean ${\displaystyle \mu }$ and unit variance (i.e., ${\displaystyle x[n]\sim {\mathcal {N}}(\mu ,1)}$).

The sample mean, ${\displaystyle {\overline {x}}}$, of the sample ${\displaystyle x[0],x[1],\ldots ,x[N-1]}$, defined as

${\displaystyle {\overline {x}}={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]}$

has variance ${\displaystyle {\frac {1}{N}}}$. This is equal to the reciprocal of the Fisher information from the sample (this is clear from the definition) and thus, by the Cramér-Rao inequality, the sample mean is efficient in the sense that its efficiency is unity.

Now consider the sample median. This is an unbiased and consistent estimator for ${\displaystyle \mu }$. For large ${\displaystyle N}$ the sample median is approximately normally distributed with mean ${\displaystyle \mu }$ and variance ${\displaystyle {\frac {\pi }{2N}}}$ (i.e., ${\displaystyle x[n]\sim {\mathcal {N}}\left(\mu ,{\frac {\pi }{2N}}\right)}$). The efficiency is thus ${\displaystyle {\frac {2}{\pi }}}$, or about 64%. Note that this is the asymptotic efficiency — that is, the efficiency in the limit as sample size ${\displaystyle N}$ tends to infinity. For finite values of ${\displaystyle N}$ the efficiency is higher than this (for example, a sample size of 3 gives an efficiency of about 74%).

Many workers prefer the sample median as an estimator of the mean, holding that the loss in efficiency is more than compensated for by its enhanced robustness in terms of its insensitivity to outliers.

Relative efficiency

If ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are estimators for the parameter ${\displaystyle \theta }$, then ${\displaystyle T_{1}}$ is said to dominate ${\displaystyle T_{2}}$ if:

1. its mean squared error (MSE) is smaller for at least some value of ${\displaystyle \theta }$
2. the MSE does not exceed that of ${\displaystyle T_{2}}$ for any value of θ.

Formally,

${\displaystyle \mathrm {E} \left[(T_{1}-\theta )^{2}\right]\leq \mathrm {E} \left[(T_{2}-\theta )^{2}\right]}$

holds for all ${\displaystyle \theta }$, with strict inequality holding somewhere.

The relative efficiency is defined as

${\displaystyle e(T_{1},T_{2})={\frac {\mathrm {E} \left[(T_{2}-\theta )^{2}\right]}{\mathrm {E} \left[(T_{1}-\theta )^{2}\right]}}}$

Although ${\displaystyle e}$ is in general a function of ${\displaystyle \theta }$, in many cases the dependence drops out; if this is so, ${\displaystyle e}$ being greater than one would indicate that ${\displaystyle T_{1}}$ is preferable, whatever the true value of ${\displaystyle \theta }$.de:Effizienz (Statistik) it:Efficienza (statistica)