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.

- Estimators
- Efficient estimators
- Asymptotic efficiency
- Dominant estimators
- Relative efficiency
- Robustness
- Uses of inefficient estimators
- Efficiency in Statistics
- Hypothesis tests
- Experimental design
- See also
- Notes
- References
- Further reading

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] }

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

where 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.

An **efficient estimator** is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. The most common choice of the loss function is quadratic, resulting in the mean squared error criterion of optimality.^{ [4] }

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 .

Here,

Therefore, an estimator T_{1} performs better than an estimator T_{2} if .^{ [5] }

For a more specific case, if T_{1} and T_{2} are two *unbiased* estimators for the same parameter θ, then the variance can be compared to determine performance.

T_{2} is *more efficient* than T_{1} if the variance of T_{2} is *smaller* than the variance of T_{1}, i.e. 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, .

Therefore, as the term drops out from being equal to 0.^{ [5] }

If an unbiased estimator of a parameter *θ* attains 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.

Suppose { *P _{θ}* |

where is the Fisher information matrix of the model at point *θ*. Generally, the variance measures the degree of dispersion of a random variable around its mean. Thus estimators with small variances are more concentrated, they estimate the parameters more precisely. We say that the estimator is a **finite-sample efficient estimator** (in the class of unbiased estimators) if it reaches the lower bound in the Cramér–Rao inequality above, for all *θ* ∈ Θ. Efficient estimators are always minimum variance unbiased estimators. However the converse is false: There exist point-estimation problems for which the minimum-variance mean-unbiased estimator is inefficient.^{ [6] }

Historically, finite-sample efficiency was an early optimality criterion. However this criterion has some limitations:

- Finite-sample efficient estimators are extremely rare. In fact, it was proved that efficient estimation is possible only in an exponential family, and only for the natural parameters of that family.
^{[ citation needed ]} - This notion of efficiency is sometimes restricted to the class of unbiased estimators. (Often it isn't.
^{ [7] }) Since there are no good theoretical reasons to require that estimators are unbiased, this restriction is inconvenient. In fact, if we use mean squared error as a selection criterion, many biased estimators will slightly outperform the “best” unbiased ones. For example, in multivariate statistics for dimension three or more, the mean-unbiased estimator, sample mean, is inadmissible: Regardless of the outcome, its performance is worse than for example the James–Stein estimator.^{[ citation needed ]} - Finite-sample efficiency is based on the variance, as a criterion according to which the estimators are judged. A more general approach is to use loss functions other than quadratic ones, in which case the finite-sample efficiency can no longer be formulated.
^{[ citation needed ]}^{[ dubious – discuss ]}

As an example, among the models encountered in practice, efficient estimators exist for: the mean *μ* of the normal distribution (but not the variance *σ*^{2}), parameter *λ* of the Poisson distribution, the probability *p* in the binomial or multinomial distribution.

Consider the model of a normal distribution with unknown mean but known variance: { *P _{θ}* =

This estimator has mean *θ* and variance of *σ*^{2} / *n*, which is equal to the reciprocal of the Fisher information from the sample. Thus, the sample mean is a finite-sample efficient estimator for the mean of the normal distribution.

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.

Consider a sample of size drawn from a normal distribution of mean and unit variance, i.e.,

The sample mean, , of the sample , defined as

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, . This is an unbiased and consistent estimator for . For large the sample median is approximately normally distributed with mean and variance ^{ [8] }

The efficiency of the median for large is thus

In other words, the relative variance of the median will be , or 57% greater than the variance of the mean – the standard error of the median will be 25% greater than that of the mean.^{ [9] }

Note that this is the asymptotic efficiency — that is, the efficiency in the limit as sample size tends to infinity. For finite values of the efficiency is higher than this (for example, a sample size of 3 gives an efficiency of about 74%).^{[ citation needed ]}

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).

If and are estimators for the parameter , then is said to ** dominate ** if:

- its mean squared error (MSE) is smaller for at least some value of
- the MSE does not exceed that of for any value of θ.

Formally, dominates if

holds for all , with strict inequality holding somewhere.

The relative efficiency of two estimators is defined as^{ [10] }

Although is in general a function of , in many cases the dependence drops out; if this is so, being greater than one would indicate that is preferable, whatever the true value of .

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.

If and are estimators for the parameter , then is said to ** dominate ** if:

- its mean squared error (MSE) is smaller for at least some value of
- the MSE does not exceed that of for any value of θ.

Formally, dominates if

holds for all , with strict inequality holding somewhere.

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.,^{ [11] }

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:

Now because we have , so the relative efficiency expresses the relative sample size of the first estimator needed to match the variance of the second.

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,* 10*σ*), 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.

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 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.

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.^{ [12] }

Pitman efficiency ^{ [13] } and Bahadur efficiency (or Hodges–Lehmann efficiency)^{ [14] }^{ [15] } relate to the comparison of the performance of statistical hypothesis testing procedures. The Encyclopedia of Mathematics provides a brief exposition of these three criteria.

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.^{ [16] }

- 1 2 Everitt 2002, p. 128.
- ↑ Nikulin, M.S. (2001) [1994], "Efficiency of a statistical procedure",
*Encyclopedia of Mathematics*, EMS Press - 1 2 Fisher, R (1921). "On the Mathematical Foundations of Theoretical Statistics".
*Philosophical Transactions of the Royal Society of London A*.**222**: 309–368. JSTOR 91208. - ↑ Everitt, B.S. (2002).
*The Cambridge Dictionary of Statistics*(2nd ed.). New York, Cambridge University Press. p. 128. ISBN 0-521-81099-X. - 1 2 Dekking, F.M. (2007).
*A Modern Introduction to Probability and Statistics: Understanding Why and How*. Springer. pp. 303-305. ISBN 978-1852338961. - ↑ Romano, Joseph P.; Siegel, Andrew F. (1986).
*Counterexamples in Probability and Statistics*. Chapman and Hall. p. 194. - ↑ DeGroot; Schervish (2002).
*Probability and Statistics*(3rd ed.). pp. 440–441. - ↑ Williams, D. (2001).
*Weighing the Odds*. Cambridge University Press. p. 165. ISBN 052100618X. - ↑ 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. - ↑ 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. - ↑ Grubbs, Frank (1965).
*Statistical Measures of Accuracy for Riflemen and Missile Engineers*. pp. 26–27. - ↑ Everitt 2002, p. 321.
- ↑ Nikitin, Ya.Yu. (2001) [1994], "Efficiency, asymptotic",
*Encyclopedia of Mathematics*, EMS Press - ↑ Arcones M. A. "Bahadur efficiency of the likelihood ratio test" preprint
- ↑ Canay I. A. & Otsu, T. "Hodges–Lehmann Optimality for Testing Moment Condition Models"
- ↑ Dodge, Y. (2006).
*The Oxford Dictionary of Statistical Terms*. Oxford University Press. ISBN 0-19-920613-9.

In statistics, an **estimator** is a rule for calculating an estimate of a given quantity based on observed data: thus the rule, the quantity of interest and its result are distinguished.

In statistics, **maximum likelihood estimation** (**MLE**) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

In statistics, the **mean squared error** (**MSE**) or **mean squared deviation** (**MSD**) of an estimator measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.

In statistics, the **Rao–Blackwell theorem**, sometimes referred to as the **Rao–Blackwell–Kolmogorov theorem**, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.

In estimation theory and statistics, the **Cramér–Rao bound (CRB)** expresses a lower bound on the variance of unbiased estimators of a deterministic parameter, stating that the variance of any such estimator is at least as high as the inverse of the Fisher information. The result is named in honor of Harald Cramér and C. R. Rao, but has independently also been derived by Maurice Fréchet, Georges Darmois, as well as Alexander Aitken and Harold Silverstone.

In mathematical statistics, the **Fisher information** is a way of measuring the amount of information that an observable random variable *X* carries about an unknown parameter *θ* of a distribution that models *X*. Formally, it is the variance of the score, or the expected value of the observed information. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher. The Fisher information is also used in the calculation of the Jeffreys prior, which is used in Bayesian statistics.

In statistics, a **consistent estimator** or **asymptotically consistent estimator** is an estimator—a rule for computing estimates of a parameter *θ*_{0}—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to *θ*_{0}. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to *θ*_{0} converges to one.

**Estimation theory** is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

In statistics a **minimum-variance unbiased estimator (MVUE)** or **uniformly minimum-variance unbiased estimator (UMVUE)** is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

In statistics, **ordinary least squares** (**OLS**) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function.

In probability theory and directional statistics, the **von Mises distribution** is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the *N*-dimensional sphere.

In statistics, an **empirical distribution function** is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/*n* at each of the *n* data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.

In statistics, a **pivotal quantity** or **pivot** is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters. A pivot quantity need not be a statistic—the function and its *value* can depend on the parameters of the model, but its *distribution* must not. If it is a statistic, then it is known as an *ancillary statistic.*

In estimation theory and decision theory, a **Bayes estimator** or a **Bayes action** is an estimator or decision rule that minimizes the posterior expected value of a loss function. Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.

In statistics, the **bias** of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called **unbiased**. In statistics, "bias" is an **objective** property of an estimator. Bias can also be measured with respect to the median, rather than the mean, in which case one distinguishes *median*-unbiased from the usual *mean*-unbiasedness property. Bias is a distinct concept from consistency. Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more.

In statistics and in particular statistical theory, **unbiased estimation of a standard deviation** is the calculation from a statistical sample of an estimated value of the standard deviation of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis.

In statistics, **Bessel's correction** is the use of *n* − 1 instead of *n* in the formula for the sample variance and sample standard deviation, where *n* is the number of observations in a sample. This method corrects the bias in the estimation of the population variance. It also partially corrects the bias in the estimation of the population standard deviation. However, the correction often increases the mean squared error in these estimations. This technique is named after Friedrich Bessel.

In statistics, **Fisher consistency**, named after Ronald Fisher, is a desirable property of an estimator asserting that if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained.

In statistics, **maximum spacing estimation**, or **maximum product of spacing estimation (MPS)**, is a method for estimating the parameters of a univariate statistical model. The method requires maximization of the geometric mean of *spacings* in the data, which are the differences between the values of the cumulative distribution function at neighbouring data points.

In statistics, the **variance function** is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function plays a large role in many settings of statistical modelling. It is a main ingredient in the generalized linear model framework and a tool used in non-parametric regression, semiparametric regression and functional data analysis. In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity. In a non-parametric setting, the variance function is assumed to be a smooth function.

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

- Lehmann, E.L.; Casella, G. (1998).
*Theory of Point Estimation*(2nd ed.). Springer. ISBN 0-387-98502-6. - Pfanzagl, Johann; with the assistance of R. Hamböker (1994).
*Parametric Statistical Theory*. Berlin: Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.

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