  The advent of large-scale inference has spurred reexamination of conventional
statistical thinking. In a Gaussian model for $n$ many $z$-scores with at most
$k < \frac{n}{2}$ nonnulls, Efron suggests estimating the location and scale
parameters of the null distribution. Placing no assumptions on the nonnull
effects, the statistical task can be viewed as a robust estimation problem.
However, the best known robust estimators fail to be consistent in the regime
$k \asymp n$ which is especially relevant in large-scale inference. The failure
of estimators which are minimax rate-optimal with respect to other formulations
of robustness (e.g. Huber's contamination model) might suggest the
impossibility of consistent estimation in this regime and, consequently, a
major weakness of Efron's suggestion. A sound evaluation of Efron's model thus
requires a complete understanding of consistency. We sharply characterize the
regime of $k$ for which consistent estimation is possible and further establish
the minimax estimation rates. It is shown consistent estimation of the location
parameter is possible if and only if $\frac{n}{2} - k = \omega(\sqrt{n})$, and
consistent estimation of the scale parameter is possible in the entire regime
$k < \frac{n}{2}$. Faster rates than those in Huber's contamination model are
achievable by exploiting the Gaussian character of the data. The minimax upper
bound is obtained by considering estimators based on the empirical
characteristic function. The minimax lower bound involves constructing two
marginal distributions whose characteristic functions match on a wide interval
containing zero. The construction notably differs from those in the literature
by sharply capturing a scaling of $n-2k$ in the minimax estimation rate of the
location.
