Best linear unbiased prediction ($\hat{b}=D{Z}^T{V}^{-1}(Y-X{\hat{\beta }}_{\mathrm{GLS}})$)

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 $D{Z}^T{V}^{-1}(Y-X{\hat{\beta }}_{\mathrm{GLS}})$, 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.