en>Kcrca: /* Analogy to the estimated final score of a cricket batsman */ Tried to make more accessible to non-cricket-knowing folk

2015-01-06T05:19:51Z

Analogy to the estimated final score of a cricket batsman: Tried to make more accessible to non-cricket-knowing folk

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	In [[statistics]], 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 loss function\|quadratic]], resulting in the [[mean squared error]] criterion of optimality.<ref>{{harvtxt\|Everitt\|2002\|p=128}}</~~ref>~~

	~~== Finite-sample efficiency ==~~
	~~Suppose {{nowrap\|{ ''P<sub>θ<~~/~~sub>'' {{!}} ''θ'' ∈ Θ }}} is a [[parametric model]] and {{nowrap\|1=''X'' = (''X''<sub>1</sub>, …, ''X<sub>n</sub>'')}} is the data sampled from this model~~. ~~Let {{nowrap\|1=''T'' = ''T''(''X'')}} be the [[estimator]] for the parameter ''θ''~~. ~~If this estimator is [[bias of an estimator\|unbiased]] (that is, {{nowrap\|1=E[ ''T'' ]~~ = ~~''θ''}}), then the [[Cramér–Rao inequality~~]~~] states the [[variance]] of this estimator~~ is ~~bounded from below:~~
	~~: <math>~~
	~~\operatorname{Var}[\,T\,]\ \geq\ \mathcal{I}_\theta^{-1},~~
	~~</math>~~
	~~where <math>\scriptstyle\mathcal{I}_\theta</math> 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 '''finite-sample efficient estimator''' (in the class of unbiased estimators) if it reaches the lower bound in the Cramér–Rao inequality above, for all {{nowrap\|''θ'' ∈ Θ}}. Efficient estimators are always [[minimum variance unbiased estimator]]s~~. However the converse is false: There exist point-estimation problems for which the minimum-variance mean-unbiased estimator is inefficient.{{Citation needed\|date=February 2012}}~~

	~~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\|date=February 2012}}
	* This notion of efficiency is restricted to ~~the class of [[bias of an estimator\|unbiased]] estimators. 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 [[admissible procedure\|inadmissible]]: Regardless of the outcome, its performance is worse than for example the [[James–Stein estimator]].~~{{Citation needed\|date=December 2011}}~~
	* 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 function]]s other than quadratic ones, in which case the finite-sample efficiency can ~~no longer~~ be ~~formulated~~.~~{{Citation needed\|date=February 2012}}{{dubious\|date=February 2012}}~~

	~~=== Example ===~~
	~~Among the models encountered in practice, efficient estimators exist~~ for: the mean ''μ'' of the [[normal distribution]] (but not the variance ''σ''<sup>2</sup>), parameter ''λ'' of the [[Poisson distribution]], the probability ''p'' in the [[binomial distribution\|binomial]] or [[multinomial distribution]].

	~~Consider the model of~~ a [[normal distribution]] with unknown mean but known variance: {{nowrap\|1={ ''P<sub>θ</sub>'' = ''N''(''θ'', ''σ''<sup>2</sup>) {{!}} ''θ'' ∈ '''R''' }.}} The data consists of ''n'' [[iid]] observations from this model: {{nowrap\|1=''X'' = (''x''<sub>1</sub>, …, ''x<sub>n</sub>'')}}. We estimate the parameter ''θ'' using the [[sample mean]] of ~~all observations:~~
	~~: <math>~~
	~~T(X) = \frac1n \sum_{i=1}^n x_i\~~ .
	~~</math>~~
	~~This estimator~~ has ~~mean ''θ''~~ and ~~variance of {{nowrap\|''σ''<sup>2</sup> / ''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.~~

	~~==Relative efficiency==~~
	~~If <math>T_1</math> and <math>T_2</math> are estimators for the parameter <math>\theta</math>, then <math>T_1</math> is said~~ to ~~'''[[dominating decision rule\|dominate]]''' <math>T_2</math> if~~:
	~~# its [[mean squared error]] (MSE) is smaller for at least some value of <math>\theta<~~/~~math>~~
	~~# the MSE does not exceed that of <math>T_2<~~/~~math> for any value of θ~~.

	~~Formally,~~ <~~math~~>~~T_1~~<~~/math~~> ~~dominates <math>T_2</math> if~~
	~~:<math>~~
	~~\mathrm{E}~~
	~~\left~~[
	~~(T_1 - \theta)^2~~
	~~\right]~~
	~~\leq~~
	~~\mathrm{E}~~
	~~\left[~~
	~~(T_2-\theta)^2~~
	~~\right]~~
	~~</math>~~

	~~holds for all <math>\theta</math>, with strict inequality holding somewhere.~~

	~~The relative efficiency is defined as~~

	:~~<math>~~
	~~e(T_1,T_2)~~
	=
	~~\frac~~
	~~{\mathrm{E} \left[ (T_2-\theta)^2 \right]}~~
	~~{\mathrm{E} \left[ (T_1-\theta)^2 \right]}~~
	~~</math>~~

	~~Although <math>e<~~/~~math> is in general a function of <math>\theta<~~/~~math>, in many cases the dependence drops out; if this is so, <math>e</math> being greater than one would indicate that <math>T_1</math> is preferable, whatever the true value of <math>\theta</math>.~~

	~~==Asymptotic efficiency==~~
	~~For some [[estimator]]s, 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.~~

	~~==See also==~~
	* [[Bayes estimator]]
	*[[Hodges’ estimator]]
	*[[Efficiency (statistics)]]

	~~==Notes==~~
	~~{{Reflist\|3}}~~

	~~==References==~~
	~~{{refbegin}}~~
	*{{cite book
	~~\| last = Everitt \| first = B.S.~~
	~~\| year = 2002~~
	~~\| title = The Cambridge Dictionary of Statistics~~
	~~\| publisher = New York, Cambridge University Press~~
	~~\| edition = 2nd~~
	~~\| isbn = 0-521-81099-X~~
	~~\| ref = harv~~
	}}
	~~{{refend}}~~

	~~==Further reading==~~
	* {{cite book
	~~\| last1 = Lehmann \| first1 = E~~.~~L. \| authorlink = Erich Leo Lehmann~~
	~~\| last2 = Casella \| first2 = G.~~
	~~\| title = Theory~~ of ~~Point Estimation, 2nd ed~~
	~~\| year = 1998~~
	~~\| publisher = Springer~~
	~~\| isbn = 0-387-98502-6~~
	}}
	* {{cite book\|title=Parametric Statistical Theory \| last1=Pfanzagl \| first1=Johann \|authorlink=Johann Pfanzagl \|last2=with the assistance of R. ~~Hamböker \|year=1994\|publisher=Walter de Gruyter\|location=Berlin\|isbn=~~3~~-11-013863-8\| mr=1291393 }}~~

	~~{{DEFAULTSORT:Efficient Estimator}}~~
	~~[[Category:Estimation theory]]~~
	~~[[Category:Statistical theory]]~~
	~~[[Category:Statistical terminology]~~]

192.76.7.203: /* See also */

2014-02-01T14:42:15Z

en>R'n'B: Fix links to disambiguation page Infinite

2012-08-23T00:19:21Z

Fix links to disambiguation page Infinite

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en>Kcrca: /* Analogy to the estimated final score of a cricket batsman */ Tried to make more accessible to non-cricket-knowing folk

192.76.7.203: /* See also */

en>R'n'B: Fix links to disambiguation page Infinite