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Marginal association summary statistics have attracted great attention instatistical genetics, mainly because the primary results of most genome-wideassociation studies (GWAS) are produced by marginal screening. In this paper,we study the prediction accuracy of marginal estimator in dense (or sparsityfree) high-dimensional settings with $(n,p,m) \to \infty$, $m/n \to \gamma \in(0,\infty)$, and $p/n \to \omega \in (0,\infty)$. We consider a generalcorrelation structure among the $p$ features and allow an unknown subset $m$ ofthem to be signals. As the marginal estimator can be viewed as a ridgeestimator with regularization parameter $\lambda \to \infty$, we furtherinvestigate a class of ridge-type estimators in a unifying framework, includingthe popular best linear unbiased prediction (BLUP) in genetics. We find thatthe influence of $\lambda$ on out-of-sample prediction accuracy heavily dependson $\omega$. Though selecting an optimal $\lambda$ can be important when $p$and $n$ are comparable, it turns out that the out-of-sample $R^2$ of ridge-typeestimators becomes near-optimal for any $\lambda \in (0,\infty)$ as $\omega$increases. For example, when features are independent, the out-of-sample $R^2$is always bounded by $1/\omega$ from above and is largely invariant to$\lambda$ given large $\omega$ (say, $\omega>5$). We also find that in-sample$R^2$ has completely different patterns and depends much more on $\lambda$ thanout-of-sample $R^2$. In practice, our analysis delivers useful messages forgenome-wide polygenic risk prediction and computation-accuracy trade-off indense high-dimensions. We numerically illustrate our results in simulationstudies and a real data example.

Cross-trait prediction accuracy of high-dimensional ridge-type estimators in genome-wide association studies