We consider the problem of approximating the empirical Shannon entropy of a
high-frequency data stream when space limitations make exact computation
infeasible. It is known that \alpha-dependent quantities such as the Renyi and
Tsallis entropies can be estimated efficiently and unbiasedly from
low-dimensional \alpha-stable data sketches. An approximation to the Shannon
entropy can be obtained from either of these quantities by taking \alpha
sufficiently close to 1. However, practical guidelines for the choice of
$\alpha$ are lacking. We avoid this problem by going directly to the limit. We
show that the projection variables used in estimating the Renyi entropy can be
transformed to have a proper distributional limit as \alpha approaches 1. The
Shannon entropy can then be estimated directly from a data sketch based on this
limiting distribution. We derive properties of the distribution, showing that
it has a surprisingly simple characteristic function (i \theta)^{i \theta} and
that the $k$th moment of the exponential of such a variable is $k^k$ for all
non-negative real values of k. These properties enable the Shannon entropy to
be estimated directly from the associated data sketch as the logarithm of a
simple average. We obtain the Fisher information for the statistical problem of
recovering the entropy from the data sketch and hence a lower bound on the
standard error of the estimated entropy. We show that our proposed estimator
has theoretical statistical efficiency of 96.8% and confirm this with an
empirical study. Finally we demonstrate that in order for the estimator to have
1+\epsilon coverage with high probability the sketch must have size
O(1/\epsilon^2), in agreement with theoretical bounds.