Best linear unbiased prediction

Suppose that the model for observations {Yj ; j = 1, ..., n} is written as :Y_j= \mu + x_j^T\beta +\xi_j + \varepsilon_j, \, where \mu is the mean of all observations Y, and ξj and εj represent the random effect and observation error for observation j, and suppose they are uncorrelated and have known variances σξ2 and σε2, respectively. Further, xj is a vector of independent variables for the jth observation and \beta is a vector of regression parameters. The BLUP problem of providing an estimate of the observation-error-free value for the kth observation, :\widetilde{Y}_k= \mu + x_k^T\beta +\xi_k , can be formulated as requiring that the coefficients of a linear predictor, defined as :\widehat{Y}_k= \sum_{j=1}^n c_{j,k} Y_j , should be chosen so as to minimise the variance of the prediction error, :V= \operatorname{Var}(\widetilde{Y}_k- \widehat{Y}_k), subject to the condition that the predictor is unbiased, :\operatorname{E}(\widetilde{Y}_k- \widehat{Y}_k)=0 . ==BLUP vs BLUE==

In contrast to the case of best linear unbiased estimation, the "quantity to be estimated", \widetilde{Y}_k, not only has a contribution from a random element but one of the observed quantities, specifically Y_k which contributes to \widehat{Y}_k, also has a contribution from this same random element. In contrast to BLUE, BLUP takes into account known or estimated variances. ==History of BLUP in breeding==

Henderson explored breeding from a statistical point of view. His work assisted the development of the selection index (SI) and estimated breeding value (EBV). These statistical methods influenced the artificial insemination stud rankings used in the United States. These early statistical methods are confused with the BLUP now common in livestock breeding. The actual term BLUP originated out of work at the University of Guelph in Canada by Daniel Sorensen and Brian Kennedy, in which they extended Henderson's results to a model that includes several cycles of selection. This model was popularized by the University of Guelph in the dairy industry under the name BLUP. Further work by the University showed BLUP's superiority over EBV and SI leading to it becoming the primary genetic predictor. There is thus confusion between the BLUP model popularized above with the best linear unbiased prediction statistical method which was too theoretical for general use. The model was supplied for use on computers to farmers. In Canada, all dairies report nationally. The genetics in Canada were shared making it the largest genetic pool and thus source of improvements. This and BLUP drove a rapid increase in Holstein cattle quality. ==See also==