Efficiency (statistics)

In the comparison of various statistical procedures, efficiency is a measure of quality of an estimator, of an experimental design,[1] or of a hypothesis testing procedure.[2] Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance. This article primarily deals with efficiency of estimators.

The relative efficiency of two procedures is the ratio of their efficiencies, although often this concept is used where the comparison is made between a given procedure and a notional "best possible" procedure. The efficiencies and the relative efficiency of two procedures theoretically depend on the sample size available for the given procedure, but it is often possible to use the asymptotic relative efficiency (defined as the limit of the relative efficiencies as the sample size grows) as the principal comparison measure.

An efficient estimator is characterized by a small variance or mean square error, indicating that there is a small deviance between the estimated value and the "true" value. [1]

Estimators

The efficiency of an unbiased estimator, T, of a parameter θ is defined as [3]

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

where ${\displaystyle {\mathcal {I}}(\theta )}$ is the Fisher information of the sample. Thus 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 e(T) ≤ 1.

Efficient estimators

Main article: Efficient estimator

In general, the spread of an estimator around the parameter θ is a measure of estimator efficiency and performance. This performance can be calculated by finding the mean squared error:

Let T be an estimator for the parameter θ. The mean squared error of T is the value ${\displaystyle MSE(T)=E[(T-\theta )^{2}]}$.

Here,

$${\displaystyle MSE(T)=E[(T-\theta )^{2}]=E[(T-E[T]+E[T]-\theta )^{2}]=E[(T-E[T])^{2}]+2E[T-E[T]](E[T]-\theta )+(E[T]-\theta ))^{2}=Var(T)+(E[T]-\theta )^{2}}$$

Therefore, an estimator T₁ performs better than an estimator T₂ if ${\displaystyle MSE(T_{1})<MSE(T_{2})}$.[4]

For a more specific case, if T₁ and T₂ are two unbiased estimators for the same parameter θ, then the variance can be compared to determine performance.

T₂ is more efficient than T₁ if the variance of T₂ is smaller than the variance of T₁, i.e. ${\displaystyle Var(T_{1})>Var(T_{2})}$ for all values of θ.

This relationship can be determined by simplifying the more general case above for mean squared error. Since the expected value of an unbiased estimator is equal to the parameter value, ${\displaystyle E[T]=\theta }$.

Therefore, ${\displaystyle MSE(T)=Var(T)}$ as the ${\displaystyle (E[T]-\theta )^{2}}$ term drops out from being equal to 0.[4]

If an unbiased estimator of a parameter θ attains ${\displaystyle e(T)=1}$ for all values of the parameter, then the estimator is called efficient.[3]

Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ. The Cramér–Rao lower bound is a lower bound of the variance of an unbiased estimator, representing the "best" an unbiased estimator can be.

An efficient estimator is also the minimum variance unbiased estimator (MVUE). 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 the MVUE). The MVUE estimator, even if it exists, is not necessarily efficient, because "minimum" does not mean equality holds on the Cramér–Rao inequality.

Thus an efficient estimator need not exist, but if it does, it is the MVUE.

Asymptotic efficiency

Some estimators 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.

Example: Median

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 ${\displaystyle X_{1},X_{2},\ldots ,X_{N}}$, defined as

${\displaystyle {\overline {X}}={\frac {1}{N}}\sum _{n=1}^{N}X_{n}\sim {\mathcal {N}}\left(\mu ,{\frac {1}{N}}\right).}$

The variance of the mean, 1/N (the square of the standard error) is equal to the reciprocal of the Fisher information from the sample and thus, by the Cramér–Rao inequality, the sample mean is efficient in the sense that its efficiency is unity (100%).

Now consider the sample median, ${\displaystyle {\widetilde {X}}}$. 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 ${\displaystyle {\pi }/{2N},}$[5]

${\displaystyle {\widetilde {X}}\sim {\mathcal {N}}\left(\mu ,{\frac {\pi }{2N}}\right).}$

The efficiency of the median for large ${\displaystyle N}$ is thus

${\displaystyle e\left({\widetilde {X}}\right)=\left({\frac {1}{N}}\right)\left({\frac {\pi }{2N}}\right)^{-1}=2/\pi \approx 0.64.}$

In other words, the relative variance of the median will be ${\displaystyle \pi /2\approx 1.57}$, or 57% greater than the variance of the mean – the standard error of the median will be 25% greater than that of the mean.[6]

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%).

The sample mean is thus more efficient than the sample median in this example. However, there may be measures by which the median performs better. For example, the median is far more robust to outliers, so that if the Gaussian model is questionable or approximate, there may advantages to using the median (see Robust statistics).

Dominant estimators

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, ${\displaystyle T_{1}}$ dominates ${\displaystyle T_{2}}$ if

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

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

Relative efficiency

The relative efficiency of two estimators is defined as[7]

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

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 ${\displaystyle \theta }$.

An alternative to relative efficiency for comparing estimators, is the Pitman closeness criterion. This replaces the comparison of mean-squared-errors with comparing how often one estimator produces estimates closer to the true value than another estimator.

Estimators of the mean of u.i.d. variables

In estimating the mean of uncorrelated, identically distributed variables we can take advantage of the fact that the variance of the sum is the sum of the variances. In this case efficiency can be defined as the square of the coefficient of variation, i.e.,[8]

${\displaystyle e\equiv \left({\frac {\sigma }{\mu }}\right)^{2}}$

NOTE: is this inverted? Needs explanation.

Relative efficiency of two such estimators can thus be interpreted as the relative sample size of one required to achieve the certainty of the other. Proof:

${\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {s_{1}^{2}}{s_{2}^{2}}}.}$

Now because ${\displaystyle s_{1}^{2}=n_{1}\sigma ^{2},\,s_{2}^{2}=n_{2}\sigma ^{2}}$ we have ${\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {n_{1}}{n_{2}}}}$, so the relative efficiency expresses the relative sample size of the first estimator needed to match the variance of the second.

Robustness

Efficiency of an estimator may change significantly if the distribution changes, often dropping. This is one of the motivations of robust statistics – an estimator such as the sample mean is an efficient estimator of the population mean of a normal distribution, for example, but can be an inefficient estimator of a mixture distribution of two normal distributions with the same mean and different variances. For example, if a distribution is a combination of 98% N(μ, σ) and 2% N(μ, 10σ), the presence of extreme values from the latter distribution (often "contaminating outliers") significantly reduces the efficiency of the sample mean as an estimator of μ. By contrast, the trimmed mean is less efficient for a normal distribution, but is more robust (less affected) by changes in distribution, and thus may be more efficient for a mixture distribution. Similarly, the shape of a distribution, such as skewness or heavy tails, can significantly reduce the efficiency of estimators that assume a symmetric distribution or thin tails.

Uses of inefficient estimators

Further information: L-estimator § Applications

While efficiency is a desirable quality of an estimator, it must be weighed against other considerations, and an estimator that is efficient for certain distributions may well be inefficient for other distributions. Most significantly, estimators that are efficient for clean data from a simple distribution, such as the normal distribution (which is symmetric, unimodal, and has thin tails) may not be robust to contamination by outliers, and may be inefficient for more complicated distributions. In robust statistics, more importance is placed on robustness and applicability to a wide variety of distributions, rather than efficiency on a single distribution. M-estimators are a general class of solutions motivated by these concerns, yielding both robustness and high relative efficiency, though possibly lower efficiency than traditional estimators for some cases. These are potentially very computationally complicated, however.

A more traditional alternative are L-estimators, which are very simple statistics that are easy to compute and interpret, in many cases robust, and often sufficiently efficient for initial estimates. See applications of L-estimators for further discussion.

Efficiency in Statistics

Efficiency in statistics is important because they allow one to compare the performance of various estimators. Although an unbiased estimator is usually favored over a biased one, a more efficient biased estimator can sometimes be more valuable than a less efficient unbiased estimator. For example, this can occur when the values of the biased estimator gathers around a number closer to the true value. Thus, estimator performance can be predicted easily by comparing their mean squared errors or variances.

Hypothesis tests

For comparing significance tests, a meaningful measure of efficiency can be defined based on the sample size required for the test to achieve a given task power.[9]

Pitman efficiency[10] and Bahadur efficiency (or Hodges–Lehmann efficiency)[11][12] relate to the comparison of the performance of statistical hypothesis testing procedures. The Encyclopedia of Mathematics provides a brief exposition of these three criteria.

Experimental design

Further information: Optimal design

For experimental designs, efficiency relates to the ability of a design to achieve the objective of the study with minimal expenditure of resources such as time and money. In simple cases, the relative efficiency of designs can be expressed as the ratio of the sample sizes required to achieve a given objective.[13]

Notes

1. Everitt 2002, p. 128.
2. Nikulin, M.S. (2001) [1994], "Efficiency of a statistical procedure", Encyclopedia of Mathematics, EMS Press
3. Fisher, R (1921). "On the Mathematical Foundations of Theoretical Statistics". Philosophical Transactions of the Royal Society of London A. 222: 309–368. JSTOR 91208.
4. Dekking, F.M. (2007). A Modern Introduction to Probability and Statistics: Understanding Why and How. Springer. pp. 303-305. ISBN 978-1852338961.
5. Williams, D. (2001). Weighing the Odds. Cambridge University Press. p. 165. ISBN 052100618X.
6. Maindonald, John; Braun, W. John (2010-05-06). Data Analysis and Graphics Using R: An Example-Based Approach. Cambridge University Press. p. 104. ISBN 978-1-139-48667-5.
7. Wackerly, Dennis D.; Mendenhall, William; Scheaffer, Richard L. (2008). Mathematical statistics with applications (Seventh ed.). Belmont, CA: Thomson Brooks/Cole. p. 445. ISBN 9780495110811. OCLC 183886598.
8. Grubbs, Frank (1965). Statistical Measures of Accuracy for Riflemen and Missile Engineers. pp. 26–27.
9. Everitt 2002, p. 321.
10. Nikitin, Ya.Yu. (2001) [1994], "Efficiency, asymptotic", Encyclopedia of Mathematics, EMS Press
11. Arcones M. A. "Bahadur efficiency of the likelihood ratio test" preprint
12. Canay I. A. & Otsu, T. "Hodges–Lehmann Optimality for Testing Moment Condition Models"
13. Dodge, Y. (2006). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 0-19-920613-9.

References

• Everitt, Brian S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 0-521-81099-X.CS1 maint: ref=harv (link)
• Lehmann, Erich L. (1998). Elements of Large-Sample Theory. New York: Springer Verlag. ISBN 978-0-387-98595-4.CS1 maint: ref=harv (link)