= Best linear unbiased prediction
{title2=$\widehat b=DZ^TV^{-1}(Y-X\widehat\beta_{\mathrm{GLS}})$}
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= Best linear unbiased predictor
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= BLUP
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With known covariance parameters, best linear unbiased prediction minimizes prediction-error <variance> over linear predictors that are unbiased over both observation errors and <random effects>, for every fixed-effect value. In a <Gaussian linear mixed model>, the random-effect predictor is $DZ^TV^{-1}(Y-X\widehat\beta_{\mathrm{GLS}})$, where <generalized least squares> estimates the fixed effects. Plugging in covariance estimates gives an empirical predictor; its uncertainty must also account for estimating those covariance parameters. It differs from a <best linear unbiased estimator> of an unknown fixed coefficient.
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