Best linear unbiased prediction

Suppose that the model for observations {Y_j ; j = 1, ..., n} is written as

${\displaystyle Y_{j}=\mu +x_{j}^{T}\beta +\xi _{j}+\varepsilon _{j},\,}$

where ${\displaystyle \mu }$ is the mean of all observations ${\displaystyle Y}$, and ξ_j and ε_j represent the random effect and observation error for observation j, and suppose they are uncorrelated and have known variances σ_ξ² and σ_ε², respectively. Further, x_j is a vector of independent variables for the jth observation and ${\displaystyle \beta }$ is a vector of regression parameters.

The BLUP problem of providing an estimate of the observation-error-free value for the kth observation,

${\displaystyle {\widetilde {Y}}_{k}=\mu +x_{k}^{T}\beta +\xi _{k},}$

can be formulated as requiring that the coefficients of a linear predictor, defined as

${\displaystyle {\widehat {Y}}_{k}=\sum _{j=1}^{n}c_{j,k}Y_{j},}$

should be chosen so as to minimise the variance of the prediction error,

${\displaystyle V=\operatorname {Var} ({\widetilde {Y}}_{k}-{\widehat {Y}}_{k}),}$

subject to the condition that the predictor is unbiased,

${\displaystyle \operatorname {E} ({\widetilde {Y}}_{k}-{\widehat {Y}}_{k})=0.}$

In contrast to the case of best linear unbiased estimation, the "quantity to be estimated", ${\displaystyle {\widetilde {Y}}_{k}}$, not only has a contribution from a random element but one of the observed quantities, specifically ${\displaystyle Y_{k}}$ which contributes to ${\displaystyle {\widehat {Y}}_{k}}$, also has a contribution from this same random element.

In contrast to BLUE, BLUP takes into account known or estimated variances.[2]

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.[3] 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.

• Kriging
• Minimum mean square error

• Henderson, C.R. (1975). "Best linear unbiased estimation and prediction under a selection model". Biometrics. 31 (2): 423–447. doi:10.2307/2529430. JSTOR 2529430. PMID 1174616.
• Liu, Xu-Qing; Rong, Jian-Ying; Liu, Xiu-Ying (2008). "Best linear unbiased prediction for linear combinations in general mixed linear models". Journal of Multivariate Analysis. 99 (8): 1503–1517. doi:10.1016/j.jmva.2008.01.004.