explore the possibility of using sandwich method to obtain the prediction sd

[clarkliming]

currently in predict of prediction sd( including variance of $\beta$ and $\theta$, excluding the $\epsilon$ we are using bootstrapping method. The efficiency is low and takes very long time.

Explore the chance that we can use sandwich method to obtain the variance

some notes:

$p = E(y_1) = X_1 \beta - \Sigma_{12} \Sigma_{22}^{-1}X_2\beta + \Sigma_{12} \Sigma_{22}^{-1}Y_2$
$var(p) = E(var(p|\theta)) + var(E(p|\theta))$

$E(p|\theta) = (X_1 - \Sigma_{12} \Sigma_{22}^{-1}X_2) \beta(\theta) + \Sigma_{12} \Sigma_{22}^{-1}Y_2$
the derivatives, $\frac{\partial{Sigma}}{\partial{\theta}}$ are already obtained in KR/Satterthewaite methods.
$\beta(\theta))$ is a non-linear function of $\theta$ but can also be approximated as linear.

$var(p|\theta) = (X_1 - \Sigma_{12} \Sigma_{22}^{-1}X_2)cov(\beta) (X_1 - \Sigma_{12} \Sigma_{22}^{-1}X_2)^T $
$E(var(p|\theta)) \sim var(p|\hat\theta)$

[danielinteractive]

Thanks @clarkliming !
That could be nice. I guess as you write it could work for predict, but for simulate I think we can keep in any case the current bootstrap approach (also to make sure that predict results match the easier to debug but slow simulate)