= Conditional mode of Gaussian random effects
{title2=$\widehat b=DZ^TV^{-1}(Y-X\beta)$}
In a <Gaussian linear mixed model> with $V=ZDZ^T+R$, the <conditional multivariate normal distribution> of $b$ given $Y$ has mean $DZ^TV^{-1}(Y-X\beta)$ and <covariance matrix> $D-DZ^TV^{-1}ZD$. When nonsingular, its mean is also its mode. Estimated parameters yield empirical conditional modes and shrink group deviations toward zero. At a zero variance component the associated effect is degenerate at zero; the covariance formula still applies, whereas formulas involving $D^{-1}$ require limits.
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