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For high-dimensional linear regression models, we review and compare severalestimators of variances $\tau^2$ and $\sigma^2$ of the random slopes anderrors, respectively. These variances relate directly to ridge regressionpenalty $\lambda$ and heritability index $h^2$, often used in genetics. Directand indirect estimators of these, either based on cross-validation (CV) ormaximum marginal likelihood (MML), are also discussed. The comparisons includeseveral cases of covariate matrix $\mathbf{X}_{n \times p}$, with $p \gg n$,such as multi-collinear covariates and data-derived ones. In addition, we studyrobustness against departures from the model such as sparse instead of denseeffects and non-Gaussian errors.  An example on weight gain data with genomic covariates confirms the goodperformance of MML compared to CV. Several extensions are presented. First, tothe high-dimensional linear mixed effects model, with REML as an alternative toMML. Second, to the conjugate Bayesian setting, which proves to be a goodalternative. Third, and most prominently, to generalized linear models forwhich we derive a computationally efficient MML estimator by re-writing themarginal likelihood as an $n$-dimensional integral. For Poisson and Binomialridge regression, we demonstrate the superior accuracy of the resulting MMLestimator of $\lambda$ as compared to CV. Software is provided to enablereproduction of all results presented here.

Estimation of variance components, heritability and the ridge penalty in high-dimensional generalized linear models