Build the Kronecker-structured random-effect precision matrix

For \(\Sigma_\alpha = \Sigma_{\mathrm{left}} \otimes \Sigma_{\mathrm{right}}\) (column-stacking convention), returns $$ \sigma_\varepsilon^2 \cdot (\Sigma_{\mathrm{left}}^{-1} \otimes \Sigma_{\mathrm{right}}^{-1}) $$ which is added to crossprod(Z) inside the random-effect block of the VCMM Hessian. This is the structured analogue of (sigma_eps^2 / sigma_alpha^2) * I_q used under re_cov = "diag".

Usage

build_kronecker_precision(Sigma_left, Sigma_right, sigma_eps)

Arguments

Sigma_left

A \(k \times k\) positive-definite numeric matrix (\(k = 2\) for OD-style; arbitrary \(k\) for general separable).

Sigma_right

A \(G \times G\) positive-definite numeric matrix.

sigma_eps

Positive numeric. Residual standard deviation.

Value

A \(kG \times kG\) numeric matrix.

References

Jalili, L. and Lin, L.-H. (2025). Scalable and Communication-Efficient Varying Coefficient Mixed-Effects Models.