  We study asymptotic minimax problems for estimating a $d$-dimensional
regression parameter over spheres of growing dimension ($d\to \infty$).
Assuming that the data follows a linear model with Gaussian predictors and
errors, we show that ridge regression is asymptotically minimax and derive new
closed form expressions for its asymptotic risk under squared-error loss. The
asymptotic risk of ridge regression is closely related to the Stieltjes
transform of the Mar\v{c}enko-Pastur distribution and the spectral distribution
of the predictors from the linear model. Adaptive ridge estimators are also
proposed (which adapt to the unknown radius of the sphere) and connections with
equivariant estimation are highlighted. Our results are mostly relevant for
asymptotic settings where the number of observations, $n$, is proportional to
the number of predictors, that is, $d/n\to\rho\in(0,\infty)$.
