Moment-based estimator for the Kronecker left component
Source:R/covariance.R
estimate_kronecker_components.RdGiven 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
$$\hat\Sigma_{\mathrm{left}} =
\tfrac{1}{G}\, M^{\top}\, \Sigma_{\mathrm{right}}^{-1}\, M.$$
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 covariancecov(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.