Example
Suppose that the model for observations {Yj; j = 1, ..., n} is written as
where 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 β is a vector of regression parameters. The BLUP problem of providing an estimate of the observation-error-free value for the kth observation,
can be formulated as requiring that the coefficients of a linear predictor, defined as
should be chosen so as to minimise the variance of the prediction error,
subject to the condition that the predictor is unbiased,
In contrast to the case of best linear unbiased estimation, the "quantity to be estimated", not only has a contribution from a random element but one of the observed quantities, specifically which contributes to, also has a contribution from this same random element.
Read more about this topic: Best Linear Unbiased Prediction
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