Formualation: Consider a special linear mixed effects model expressed in two stages. In stage I, the individual model is \begin{equation} Y_i=C_i\beta_i + e_i,\, \mathbb{E}(e_i|x_i)=0,\, \mathbb{V}(e_i|x_i)=R_i \tag{1} \end{equation} for full column rank design matrix $C_i\in\mathbb{R}^{n_i\times k}$. In the second stage, the population model is \begin{equation} \beta_i=A_i\beta+b_i,\, \mathbb{E}(b_i|x_i)=0,\,\mathbb{V}(b_i|x_i)=D \tag{2} \end{equation} where $\beta \in \mathbb{R}^{p}$ is the vector of fixed effects, $A_i$ is a design matrix depending on among-individual covariates, and $b_i$ and $e_i$ are independent of each other for each $i$ and $b_i,e_i,x_i$ and thus $Y_i$ are all independent across $i$. Substituting (2) in (1) yields the linear effects model \begin{equation} Y_i=X_i\beta+C_ib_i+e_i \tag{3} \end{equation} where $X_i=C_iA_i$. Assume $R_i$ and $D$ are all positive-definite matrices.
Problem: Consider the weighted least squares estimator based on i's data only, \begin{equation} \hat{\beta}_i=(C_i^TR_i^{-1}C_i)^{-1}C_i^TR_i^{-1}Y_i \end{equation} Find the conditional variance of $\mathbb{V}(\hat{\beta}_i|x_i)$ expressd in terms of $W_i$, where $W_i^{-1}=C_i^T(R_i+C_iDC_i^T)^{-1}C_i$
Thoughts: I know how to express variance in \begin{equation} \begin{aligned} \mathbb{V}(\hat{\beta}_i|x_i)&=(C_i^TR_i^{-1}C_i)^{-1}C_i^TR_i^{-1} \mathbb{V}(C_iA_i\beta+C_ib_i+e_i) R_i^{-T}C_i(C_i^TR_i^{-1}C_i)^{-T}\\ &=(C_i^TR_i^{-1}C_i)^{-1}C_i^TR_i^{-1} (C_iDC_i^T+R_i) R_i^{-T}C_i(C_i^TR_i^{-1}C_i)^{-T}\\ &=D+(C_i^TR_i^{-1}C_i)^{-1} \end{aligned} \end{equation} but I don't know how to proceed. Any ideas? Thanks.