Datasets:

Monketoo
/

math-docs-dataset

as BLUP in the literature (McGilchrist 1993), although this modal predictor is neither linear nor unbiased in general. The orthodox BLUP of the ran- dom effects is the linear unbiased predictor of U given Y which minimizes the mean square distance between the random effects U and their predictor within the class of linear functions of Y.

Explicit expressions for the mean square distances between the compo- nents of the random effects U and their predictors are as follows:

$\begin{align} c_i &= E(\hat{U}_i - U_i)^2 \\ &= \frac{\sigma^2}{1 + \sigma^2 \sum_{s=1}^{a} \sum_{h=1}^{q_s} \sum_{(i,j,k) \in \mathcal{R}(\tau_{sh})} w_{ij} \mu_{ijk,h}^{(s)}}, \tag{12} \end{align}$

where (i, j, k) runs over the risk set $\mathcal{R}(\tau_{sh})$ for fixed i. Here,

$\begin{align*} \mu_{ijk,h}^{(s)} &= \exp (\alpha_{sh} + \beta^{\top} \mathbf{x}_{ijk}^{(s)}) \\ &= \exp ((\alpha^{\top}, \beta^{\top}) \mathbf{x}_{ijk,h}^{(s)}) \\ &= \exp (\gamma^{\top} \mathbf{x}_{ijk,h}^{(s)}), \end{align*}$

and, for fixed (i, j),

$w_{ij} = \left( 1 + \omega^2 \sum_{s=1}^{a} \sum_{h=1}^{q_s} \sum_{(i,j,k) \in \mathcal{R}(\tau_{sh})} \mu_{ijk,h}^{(s)} \right)^{-1},$

where (i, j, k) runs over the risk set $\mathcal{R}(\tau_{sh})$. Similarly, we have

$\begin{align} c_{ij} &= E(\hat{U}_{ij} - U_{ij})^2 \\ &= w_{ij} \{ \omega^2 + c_i w_{ij} \}. \tag{13} \end{align}$

The cluster random effects predictor can be expressed as

$\begin{align*} \hat{U}_i &= \frac{1 + \sigma^2 \sum_{s=1}^{a} \sum_{h=1}^{q_s} \sum_{(i,j,k) \in \mathcal{R}(\tau_{sh})} w_{ij} Y_{ijk,h}^{(s)}}{1 + \sigma^2 \sum_{s=1}^{a} \sum_{h=1}^{q_s} \sum_{(i,j,k) \in \mathcal{R}(\tau_{sh})} w_{ij} \mu_{ijk,h}^{(s)}} \\ &= c_i \left( \frac{1}{\sigma^2} + \sum_{s=1}^{a} \sum_{h=1}^{q_s} \sum_{(i,j,k) \in \mathcal{R}(\tau_{sh})} w_{ij} Y_{ijk,h}^{(s)} \right), \end{align*}$

where (i, j, k) runs over the risk set $\mathcal{R}(\tau_{sh})$ for any given i. The sub-cluster random effects predictors are

$\hat{U}_{ij} = w_{ij}\hat{U}_i + \omega^2 w_{ij} \sum_{s=1}^{a} \sum_{h=1}^{q_s} \sum_{(i,j,k) \in \mathcal{R}(\tau_{sh})} Y_{ijk,h}^{(s)},$