  Suppose the expectation $E(F(X))$ is to be estimated by the empirical
averages of the values of $F$ on independent and identically distributed
samples $\{X_i\}$. A sampling rule called the "screened" estimator is
introduced, and its performance is studied. When the mean $E(U(X))$ of a
different function $U$ is known, the estimates are "screened," in that we only
consider those which correspond to times when the empirical average of the
$\{U(X_i)\}$ is sufficiently close to its known mean. As long as $U$ dominates
$F$ appropriately, the screened estimates admit exponential error bounds, even
when $F(X)$ is heavy-tailed. The main results are several nonasymptotic,
explicit exponential bounds for the screened estimates. A geometric
interpretation, in the spirit of Sanov's theorem, is given for the fact that
the screened estimates always admit exponential error bounds, even if the
standard estimates do not. And when they do, the screened estimates' error
probability has a significantly better exponent. This implies that screening
can be interpreted as a variance reduction technique. Our main mathematical
tools come from large deviations techniques. The results are illustrated by a
detailed simulation example.
