estimate_kronecker_components: Moment-based estimator for the Kronecker left component

In cevcmm: Communication-Efficient Varying Coefficient Mixed-Effects Models

Moment-based estimator for the Kronecker left component

Description

Given a length-kG random-effects estimate \hat\alpha = \mathrm{vec}_{\mathrm{col}}(M) (column-stacked, M \in \mathbb R^{G \times k}) and the right-side covariance Sigma_right, returns the weighted moment estimate

\hat\Sigma_{\mathrm{left}} = \tfrac{1}{G}\, M^{\top}\, \Sigma_{\mathrm{right}}^{-1}\, M.

Usage

estimate_kronecker_components(alpha, n_groups, q_left = 2L, Sigma_right = NULL)

Arguments

alpha	Numeric vector of length kG.
n_groups	Integer G.
q_left	Integer k, the left (within) dimension. Defaults to 2 for backward compatibility with the OD setting.
Sigma_right	Optional G \times G positive-definite covariance. If NULL, the unweighted sample covariance cov(alpha_mat) is returned (unbiased only when rows are uncorrelated).

Details

This is unbiased under the Kronecker model \alpha \sim N(0, \Sigma_{\mathrm{left}} \otimes \Sigma_{\mathrm{right}}) when Sigma_right is correct. When \hat\alpha is the BLUP rather than the true \alpha, apply the EM-style correction in vcmm (handled automatically by fit_ss / fit_csl).

Backwards-compatible alias: if q_left = 2 (the default), this is the same estimator as the previous estimate_kronecker_components for the OD setting.

Value

A k \times k symmetric positive-definite matrix.

References

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

cevcmm documentation built on July 24, 2026, 5:07 p.m.