= Solution
For the <Gaussian linear mixed model>, temporarily regard $\beta,D,R$ as known, with $R=\sigma^2I$ here and $V=ZDZ^T+R$. The joint <multivariate normal distribution> is
$$
\begin{pmatrix}b\\Y\end{pmatrix}\sim
N\!\left(\begin{pmatrix}0\\X\beta\end{pmatrix},
\begin{pmatrix}D&DZ^T\\ZD&V\end{pmatrix}\right).
$$
By the <conditional multivariate normal distribution> formula,
$$
b\mid Y\sim N\!\left(DZ^TV^{-1}(Y-X\beta),\ D-DZ^TV^{-1}ZD\right).
$$
A nonsingular Gaussian density has its mode at its mean. Thus the <conditional mode of Gaussian random effects> equals the <conditional expectation>. Replacing the unknown parameters by fitted values gives
$$
\boxed{\widehat b=\widehat D Z^T\widehat V^{-1}(Y-X\widehat\beta).}
$$
For positive-definite $D$ and $R$, an equivalent derivation minimizes the negative conditional log density,
$$
(Y-X\beta-Zb)^TR^{-1}(Y-X\beta-Zb)+b^TD^{-1}b.
$$
Differentiation yields
$$
\boxed{\widehat b=(Z^T\widehat R^{-1}Z+\widehat D^{-1})^{-1}
Z^T\widehat R^{-1}(Y-X\widehat\beta).}
$$
The inverse-covariance penalty shrinks furnace intercepts and slopes toward the population effects. For furnace $j$, set $Z_j=(\mathbf1,t_j)$ and $D_j=\operatorname{diag}(\tau_0^2,\tau_1^2)$; the same formula provides its two-component conditional mode from that furnace's observations, conditional on the fitted common parameters. If a <variance component> is zero, the covariance form remains meaningful and its corresponding <random effect> is zero, whereas the formula with $D^{-1}$ needs a limiting interpretation. With known covariance parameters these are <best linear unbiased predictors>; plug-in variance estimates give their empirical version.
Back to article page
