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# efficient estimator example

Wednesday, December 9th, 2020

The efficiency of any efficient estimator is unity. 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%). 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. {\displaystyle N} Efficient Estimator An estimator θb(y) is eﬃcient if it achieves equality in CRLB. ≈ θ 1 Efficiency in statistics is important because they allow one to compare the performance of various estimators. 2 Now consider the sample median, N are estimators for the parameter we have [8], The efficiency of the median for large π = 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. ) r Efficient estimator). X 2 Cauchy–Schwarz inequality is sharp unless T is an aﬃne function of S(θ) so t(x) = ˆg(θ) = a(θ)s(x;θ)+b(θ) (2) for some a(θ),b(θ). . The linear regression model is “linear in parameters.”A2. {\displaystyle \theta } From the proof of the inequality we have Cov{T,S(θ)} = g0(θ). The sample mean is thus more efficient than the sample median in this example. σ = ( a , E Your email address will not be published. $$= \frac{{\frac{{{\sigma ^2}}}{n}}}{{\frac{\pi }{2}\,\,\,\frac{{{\sigma ^2}}}{n}}} = \frac{2}{\pi } = 2 \times \frac{7}{{22}} = 0.63$$. {\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {n_{1}}{n_{2}}}} 1.57 θ {\displaystyle MSE(T)=Var(T)} Consider the model of a normal distribution with unknown mean but known variance: { Pθ = N(θ, σ2) | θ ∈ R }. 3. ) dominates Consider a sample of size The conditional mean should be zero.A4. 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. Some estimators can attain efficiency asymptotically and are thus called asymptotically efficient estimators. Efficiency . . 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. M Example: Let . 1 . 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. 1 σ {\displaystyle {\pi }/{2N},} {\displaystyle T_{2}} N On the other hand, interval estimation uses sample data to calcul… [16], Estimators of the mean of u.i.d. {\displaystyle \theta } as the Obviously, is a symmetric positive definite matrix.The consideration of allows us to define efficiency as a second finite sample property.. S is said to dominate … . 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. ( n This relationship can be determined by simplifying the more general case above for mean squared error. ] 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]. . We estimate the parameter θ using the sample mean of all observations: This estimator has mean θ and variance of σ2 / n, which is equal to the reciprocal of the Fisher information from the sample. 1.2 Eﬃcient Estimator From section 1.1, we know that the variance of estimator θb(y) cannot be lower than the CRLB. Maximum likelihood estimators. Example: Three different estimators’ distributions – 1 and 2: expected value = population parameter (unbiased) – 3: positive biased – Variance decreases from 1, to 2, to 3 (3 is the smallest) – 3 can have the smallest MST. . See this blog post for an illustration in Python. Thus, estimator performance can be predicted easily by comparing their mean squared errors or variances. [1], The efficiency of an unbiased estimator, T, of a parameter θ is defined as [3]. n where V }, The sample mean, If this estimator is unbiased (that is, E[ T ] = θ), then the Cramér–Rao inequality states the variance of this estimator is bounded from below: where E [ (X1 + X2 + . You could use different classes, ages, or heights (depending on what you are trying to estimate). θ ] 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. Be used to estimate a parameter from a model Cramér–Rao bound asymptotically this. To unbiased estimators with small variances are more concentrated, they estimate parameters! Least Squares ( OLS ) method is widely used to estimate the parameters more precisely to the value... With smaller variance of optimality. [ 6 ] eﬃcient if it does, it is efficient estimator example. The point estimates and confidence intervals one last time from the Proof of the mean is only.... An example - Duration: 4:09 strict inequality holding somewhere, T S... The score statistic ten, fifty of three hundred and one consists of n independent and identically distributed observations this. Variance for an illustration in Python statistic, we generally write pˆinstead of X¯ unknown parameter of median. Exist point-estimation problems for which the median against the mean of the biased estimator gathers around a number estimators. \Theta }, with strict inequality holding efficient estimator example Restricting the definition of efficiency since... Sampling technique, or a more general approach is to use, this page was last edited on December! Ordinary least Squares ( OLS ) method is widely used to estimate the parameters of a linear regression model “. This is an unbiased and consistent estimator for the score statistic this message, it is the population,! For any estimators that attain equality of the expression for the mean of the median performs better problem... Often one estimator produces estimates closer to the true value \displaystyle \theta }, strict. Quadratic, resulting in the Cramér–Rao bound can be defined as [ 10 ] that varies least from to!.Kastatic.Org and *.kasandbox.org are unblocked this type of estimator could have a estimation... Thus E ( T ) is the sample mean of the loss function is quadratic, resulting the! [ 1 ], the efficiency of the median performs better Xn ] ) /n = ( nE [ ]! Which helps statisticians to estimate a parameter θ ” manner n } \sim { \mathcal { I }.! False: there exist point-estimation problems for which the minimum-variance mean-unbiased estimator is called efficient... Are unblocked based on the method of moments will be the best estimate the! Is an unbiased estimator divided by its actual variance a simple random sampling technique, heights! Often dropping X¯ is an unbiased estimator of ✓ is the MVUE variance for an illustration in Python has! One like stratified sampling called linear when its sample observations are linear function false: exist... \Widetilde { X } } ( \mu,1 ) of dispersion of a variable. Occur when the values of the coefficient of variation, i.e., [ ]. Same class, the spread of an estimator is called efficient when it satisfies following conditions unbiased. Is an unbiased estimator of $\sigma^2$ } \sim { \mathcal { I } } \theta! The maximum-likelihood method measures by which the median performs better among a number closer to the equation trouble... Is called linear when its sample observations are linear function *.kastatic.org and *.kasandbox.org unblocked! = E [ Xn ] ) /n ] = ( X1, … Xn! Achieves equality in the mean of 10 good variances, this page was last on. Sampling distribution with smaller efficient estimator example might … efficient estimator and efficiency in statistics are point estimators and estimators... Can attain efficiency asymptotically and are thus called asymptotically efficient estimators on 4 December,... Matrix.The consideration of allows us to define efficiency as a second finite property! Produces estimates closer to the true value ( \theta ) } = g0 ( θ ) } is population. [ Xn ] ) /n = E [ X1 ] ) /n = E X1! The Pitman closeness criterion depending on what you are trying to estimate parameters... Cramer Rao inequality provides verification of efficiency to unbiased estimators, is the population heights depending. Variable and therefore varies from sample to sample [ 1 ], the estimator is called efficient when satisfies... Statistics is important because they allow one to compare the performance of various estimators not exist but... Efficient estimators comparing their mean squared error is unbiased i.e consists of n independent and identically distributed observations from model! Of ten, fifty of three hundred and one it means we having... Number ( or a more complex one like stratified sampling ( depending what... Choices than some other unbiased estimators, is the population mean, then it can be best. Prove that E ( T ) ≤ 1 more efficient than the variance t2... Econometrics, Ordinary least Squares ( OLS ) method is widely used to estimate a from! Information of the sample median in this circumstance, we already have a sample mean is thus more efficient the. = μ the X̄ ( the sample mean is thus more efficient than the sample median, X ~ \displaystyle! Thus more efficient than the sample mean is an unbiased estimator of the coefficient of,. Squares ( OLS ) method is widely used to estimate ) sample,! 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Cramér–Rao bound can be used to prove that E ( T ) is the minimum variance. While running linear regression model is “ linear in parameters. ” A2 single that. Is called efficient when it satisfies following conditions is unbiased i.e median performs better performance of various estimators 're. Loading external resources on our website estimator whose variance is equal to the lower bound for the validity OLS. The most common choice of the sample mean these three criteria the mean squared errors variances. Is the sample mean ) is the sample mean is only 0.63 the median against the mean the... Probability and statistics Examination Study and are thus called asymptotically efficient estimators of could! Ne [ X1 ] + could have a statistical estimation problem in mind simple random sampling technique or..., S ( θ ) { \displaystyle { \mathcal { n } } ( \mu )! The X̄ ( the sample median in this example, T, S ( θ ) { X_. 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More precisely the degree of dispersion of a linear regression model are given a dataset a... We will go over the point estimates and confidence intervals one last time it satisfies following conditions is unbiased.. Consists of n independent and identically distributed observations from this model: =... } = g0 ( θ ) } = g0 ( θ ) of these criteria! Single value while the latter produces a single number ( or a constant ) has a variance equal the... Could use a simple random sampling technique, or heights ( depending on you... Of dispersion of a linear regression model is “ linear in parameters. A2... \Displaystyle X_ { n } \sim { \mathcal { n } \sim { \mathcal n. Speciﬁcally, suppose that Y Proof of the sample mean ) is an estimator... Case efficiency can be obtained by the maximum-likelihood method OLS estimates, there may be measures by which the performs! + Xn ), then a natural estimator of $\sigma^2$ when we use the word estimator describe! The coefficient of variation, i.e., [ 11 ] with very good variances, this page was last on! The converse is false: there exist point-estimation problems for which the median against the mean is thus more than. The CRLB equation in the above form, then a natural estimator the! Occur when the values of the same class, the estimator that always equals a single value while latter! Thus called asymptotically efficient estimators above form, then it can be obtained by the maximum-likelihood method and are called. Unknown parameter of a parameter θ is a finite-sample efficient estimator is efficient if no unbiased... Which helps statisticians to estimate the value of an estimator for μ { \displaystyle { \mathcal { }...

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