  Modern datasets are trending towards ever higher dimension. In response,
recent theoretical studies of covariance estimation often assume the
proportional-growth asymptotic framework, where the sample size $n$ and
dimension $p$ are comparable, with $n, p \rightarrow \infty $ and $\gamma_n =
p/n \rightarrow \gamma > 0$. Yet, many datasets -- perhaps most -- have very
different numbers of rows and columns. We consider instead the
disproportional-growth asymptotic framework, where $n, p \rightarrow \infty$
and $\gamma_n \rightarrow 0$ or $\gamma_n \rightarrow \infty$. Either
disproportional limit induces novel behavior unseen within previous
proportional and fixed-$p$ analyses. We study the spiked covariance model, with
theoretical covariance a low-rank perturbation of the identity. For each of 15
different loss functions, we exhibit in closed form new optimal shrinkage and
thresholding rules. Our optimal procedures demand extensive eigenvalue
shrinkage and offer substantial performance benefits over the standard
empirical covariance estimator.
  Practitioners may ask whether to view their data as arising within (and apply
the procedures of) the proportional or disproportional frameworks.
Conveniently, it is possible to remain {\it framework agnostic}: one unified
set of closed-form shrinkage rules (depending only on the aspect ratio
$\gamma_n$ of the given data) offers full asymptotic optimality under either
framework. At the heart of the phenomena we explore is the spiked Wigner model,
in which a low-rank matrix is perturbed by symmetric noise. Exploiting a
connection to the spiked covariance model as $\gamma_n \rightarrow 0$, we
derive optimal eigenvalue shrinkage rules for estimation of the low-rank
component, of independent and fundamental interest.
