Parameter estimation in the linear mixed effects model

Question

In Parameter estimation and inference in the linear mixed effects model, page 1923, the variance

\begin{equation} \begin{aligned} \text{var}(\tilde{u} - u) & = \sigma^2G - \text{var}(\tilde{u}) \\ & = \text{var}(u) - \text{var}(\tilde{u}), \end{aligned} \end{equation}

where $\tilde{u} = GZ^\top H^{-1}(y - X\hat{\beta})$ is the best linear unbiased predictor (BLUP) for the random effects vector $u$, where $G$ and $H$ are covariance matrices, $Z$ is a design matrix, $y$ is a vector of observations, and $\hat{\beta}$ is the maximum likelihood (ML) estimate for $\beta$.

By definition,

\begin{equation} \text{var}(\tilde{u} - u) = \text{var}(u) + \text{var}(\tilde{u}) - 2\text{cov}(\tilde{u}, u), \end{equation}

this must mean that $\text{cov}(\tilde{u}, u) = \text{var}(\tilde{u})$. How can one show that $\text{cov}(\tilde{u}, u) = \text{var}(\tilde{u})$?

Answer 1

We have that $$\mbox{cov}(u, \tilde u) = E \Bigl [ \bigl \{u - E(u) \bigr \} \, \bigl \{ \tilde u - E(\tilde u)\bigr \} \Bigr ].$$

But $E(\tilde u) = u$ and $E(u) = \tilde u$. Note that expectations are here taken with respect to the posterior of the random effects, not the prior. Hence, $$\mbox{cov}(u, \tilde u) = E \Bigl [ \bigl \{\tilde u - E(\tilde u) \bigr \} \, \bigl \{ \tilde u - E(\tilde u)\bigr \} \Bigr ] = \mbox{var}(\tilde u).$$