  Marginal association summary statistics have attracted great attention in
statistical genetics, mainly because the primary results of most genome-wide
association studies (GWAS) are produced by marginal screening. In this paper,
we study the prediction accuracy of marginal estimator in dense (or sparsity
free) 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 general
correlation structure among the $p$ features and allow an unknown subset $m$ of
them to be signals. As the marginal estimator can be viewed as a ridge
estimator with regularization parameter $\lambda \to \infty$, we further
investigate a class of ridge-type estimators in a unifying framework, including
the popular best linear unbiased prediction (BLUP) in genetics. We find that
the influence of $\lambda$ on out-of-sample prediction accuracy heavily depends
on $\omega$. 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-type
estimators 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 $\lambda$ than
out-of-sample $R^2$. In practice, our analysis delivers useful messages for
genome-wide polygenic risk prediction and computation-accuracy trade-off in
dense high-dimensions. We numerically illustrate our results in simulation
studies and a real data example.
