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In [[statistics]], '''best linear unbiased prediction''' ('''BLUP''') is used in linear [[mixed model]]s for the estimation of [[random effects model|random effects]]. BLUP was derived by [[Charles Roy Henderson]] in 1950 but the term "best linear unbiased predictor" (or "prediction") seems not to have been used until 1962.<ref name=Robinson>{{cite journal
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| last=Robinson |first=G.K.
| year = 1991
| title = That BLUP is a Good Thing: The Estimation of Random Effects
| journal = [[Statistical Science]]
| volume = 6  | issue = 1  | pages = 15–32
| doi = 10.1214/ss/1177011926
| mr=1108815
| zbl=0955.62500
| jstor=2245695
}}
</ref> "Best linear unbiased predictions" (BLUPs) of random effects are similar to best linear unbiased estimates (BLUEs) (see [[Gauss–Markov theorem]]) of fixed effects. The distinction arises because it is conventional to talk about ''estimating'' fixed effects but ''predicting'' random effects, but the two terms are otherwise equivalent. (This is a bit strange since the random effects have already been "realized" − they already exist. The use of the term "prediction" may be because in the field of animal breeding in which Henderson worked, the random effects were usually genetic merit, which could be used to predict the quality of offspring (Robinson<ref name=Robinson/> page 28). However, the equations for the "fixed" effects and for the random effects are different.
 
In practice, it is often the case that the parameters associated with the random effect(s) term(s) are unknown; these parameters are the variances of the random effects and residuals. Typically the parameters are estimated and plugged into the predictor, leading to the [[Empirical Best Linear Unbiased Predictor]] (EBLUP). Notice that by simply plugging in the estimated parameter into the predictor, additional variability is unaccounted for, leading to  overly optimistic prediction variances for the EBLUP.
 
Best linear unbiased predictions are similar to [[empirical Bayes]] estimates of random effects in linear mixed models, except that in the latter case, where weights depend on unknown values of components of variance, these unknown variances are replaced by sample-based estimates.
 
==Example==
 
Suppose that the model for observations {''Y''<sub>''j''</sub>; ''j'' = 1, ..., ''n''} is written as
 
:<math>Y_j= \mu + x_j^T\beta +\xi_j + \varepsilon_j, \, </math>
 
where ''&xi;<sub>j</sub>'' and ''&epsilon;<sub>j</sub>'' represent the random effect and observation error for observation ''j'', and suppose they are uncorrelated and have known variances ''σ<sub>ξ</sub>''<sup>2</sup> and ''σ<sub>ε</sub>''<sup>2</sup>, respectively. Further, ''x<sub>j</sub>'' is a vector of [[dependent and independent variables|independent variables]] for the ''j''th observation and ''β'' is a vector of regression parameters.  The BLUP problem of providing an estimate of the observation-error-free value for the ''k''th observation,
 
:<math>\tilde{Y_k}= \mu + x_k^T\beta +\xi_k    ,</math>
 
can be formulated as requiring that the coefficients of a linear predictor, defined as
 
:<math>\widehat{Y_k}= \sum_{j=1}^n c_{j,k} Y_j  ,</math>
 
should be chosen so as to minimise the variance of the prediction error,
 
:<math>V= \operatorname{Var}(\tilde{Y_k}- \widehat{Y_k}), </math>
 
subject to the condition that the predictor is unbiased,
 
:<math>\operatorname{E}(\tilde{Y_k}- \widehat{Y_k})=0 . </math>
 
In contrast to the case of [[Best linear unbiased estimator|best linear unbiased estimation]], the "quantity to be estimated", <math>\tilde{Y_k}</math>, not only has a contribution from a random element but one of the observed quantities, specifically <math>Y_k</math> which contributes to <math>\widehat{Y_k}</math>, also has a contribution from this same random element.
 
==See also==
*[[Minimum mean square error]]
 
==Tutorials==
* [http://www.extension.org/pages/61006/the-solcap-tomato-phenotypic-data:-estimating-heritability-and-blups-for-traits#.Ui4zjWRgYXc Estimating BLUPs and Heritability Using R]
* [http://www.extension.org/pages/68019/genomic-relationships-and-gblup Genomic Relationships and GBLUP]
* [http://www.extension.org/pages/66590/rrblup-package-in-r-for-genomewide-selection Ridge Regression, rrBLUP for Genome-Wide Selection]
==References==
<references />
* {{cite journal
|last=Henderson |first=C.R. |authorlink=Charles Roy Henderson
|year=1975
|title=Best linear unbiased estimation and prediction under a selection model
|journal=[[Biometrics (journal)|Biometrics]]
|volume=31 |issue=2 |pages=423&ndash;447
|doi=10.2307/2529430 |pmid=1174616
|jstor=2529430
}}
*Xu-Qing Liu, Jian-Ying Rong, Xiu-Ying Liu (2008). "Best linear unbiased prediction for linear combinations in general mixed linear models", Journal of Multivariate Analysis, 99 (8),1503&ndash;1517. {{doi|10.1016/j.jmva.2008.01.004}}.
 
[[Category:Statistical methods]]
[[Category:Estimation theory]]

Latest revision as of 13:19, 4 June 2014

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Yes Mister. Sometimes you get these ignorant men and women who just be sure to educate you about the air jordan Shoes game advertise you be a mess. But then you just laugh at such non sense. Anyone know enhanced.But any how.

If in comparison to look for a kind of shoes that combines comfort, style and fashion together, Nike Air Max shoes will be your most suitable option. Why are nike air max shoes so hot from a long season? The reason is that their fashion look and the total comfort for the users. It is known to us that comfort will be the first and important element to the wearers. Designer Nike is actually the form of shoes. Additionally to comfort, fashion design also is yet key point people to captivate persons and attract people's vision. It is obvious that Nike has been the well-known brand in the whole world, due to people factors.

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