Efficiency

In statistics, efficiency is a term used in the comparison of various statistical procedures and, in particular, it refers to a measure of the desirability of an estimator or of an experimental design.^[1] The relative efficiency of two procedures is the ratio their efficiencies, although often this term 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.

Efficiencies are often defined using the variance or mean square error as the measure of desirability.^[2] However, for comparing significance tests, a meaningful measure can be defined based on the sample size required for the test to achieve a given power.^[3]

Contents 1 Definition for unbiased estimators 2 Efficient estimator 3 Asymptotic efficiency 4 Example 5 Relative efficiency 6 See also 7 Notes 8 References 9 External links

Definition for unbiased estimators

The efficiency of an unbiased estimator T is defined as

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 :

Efficient estimator

If an unbiased estimator of a parameter attains 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 .

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

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.

Example

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

The sample mean, , of the sample , defined as

has variance 1 / N. This 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 μ. For large N the sample median is approximately normally distributed with mean μ and variance π / 2N, i.e., The efficiency for large N is thus 2 / π, or about 64%. Note that this is the asymptotic efficiency — that is, the efficiency in the limit as sample size N tends to infinity. For finite values of 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. 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).

Relative efficiency

If T₁ and T₂ are estimators for the parameter θ, then T₁ is said to dominate T₂ if:

1. its mean squared error (MSE) is smaller for at least some value of θ
2. the MSE does not exceed that of T₂ for any value of θ.

Formally, T₁ dominates T₂ if

holds for all θ, with strict inequality holding somewhere.

The relative efficiency is defined as

Although e is in general a function of θ, in many cases the dependence drops out; if this is so, e being greater than one would indicate that T₁ is preferable, whatever the true value of θ.

See also

• Bahadur efficiency [1]
• Pitman efficiency [2]
• Bayes estimator

Notes

1. ^ Everitt (2002) p 128
2. ^ Everitt (2002) p 128
3. ^ Everitt (2002) p 321

References

• Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-x

External links

• Nikitin, Ya.Yu. (2001), "Efficiency, asymptotic", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104